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Around the Brunn-Minkowski inequality
Andrea Colesanti
Technische Universitat Berlin - Institut fur Mathematik
January 28, 2015
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
Summary
I The Brunn-Minkowski inequality
I The isoperimetric inequality
I Infinitesimal form of Brunn-Minkowski inequality
I Inequalities of Brunn-Minkowski type
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
The Brunn-Minkowski inequality
Thm. A,B ⊂ Rn, compact; λ ∈ [0, 1]; then
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n. (BM)
I Vn = volume (Lebesgue measure);
I
(1− λ)A + λB = {(1− λ)a + λb : a ∈ A, b ∈ B}.
Equivalently: The functional V1/nn is concave in the class of
compact sets of Rn, equipped with the vector addition.
An excellent survey (much better than this talk):R. Gardner, The Brunn-Minkowski inequality, Bull. A.M.S., 2002.
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A
((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
(BM) privileges convex sets
In the inequality
Vn((1− λ)A + λB)1/n ≥ (1− λ)Vn(A)1/n + λVn(B)1/n,
equality holds iff A is convex and B is a homothetic copy of A (upto subsets of volume zero).
Why? If you plug A = B in (BM) in general you don’t get anequality, because
(1− λ)A + λA 6= A ((1− λ)A + λA ⊃ A).
But if A is convex
(1− λ)A + λA = A ∀λ ∈ [0, 1].
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
Many equivalent forms
I Classic
Vn((1−λ)A+λB)1/n ≥ (1−λ)Vn(A)1/n +λVn(B)1/n. (BM)
I Elegant
Vn(A + B)1/n ≥ Vn(A)1/n + Vn(B)1/n. (BM∗)
I Multiplicative
Vn((1− λ)A + λB) ≥ Vn(A)1−λVn(B)λ. (BM0)
I Minimal
Vn((1− λ)A + λB) ≥ min{Vn(A),Vn(B)}. (BM−∞)
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
A general fact about homogeneous functional
Let F be a real-valued functional
I defined on a convex cone C;
I α-homogeneous: F(λx) = λαF(x), ∀x ∈ C, ∀λ > 0 (α > 0);
I non-negative.
F1/α concave ⇔ {F ≥ t} is convex ∀ t.
The last condition (quasi-concavity) is equivalent to
F((1− λ)A + λB) ≥ min{F(A),F(B)} ∀A,B ∈ C, ∀λ ∈ [0, 1].
In our case: C = {compact sets}, F = Vn, α = n.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality).
Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions,
and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1].
Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
An “elementary” proof of (BM) - I
Lemma (Prekopa-Leindler inequality). Let
f , g , h : Rn → R+
be measurable functions, and let λ ∈ [0, 1]. Assume that
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
Then ∫Rn
fdz ≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ.
Proof.
I Prove the 1-dimensional case (using just the so-called layercake, or Cavalieri’s, principle);
I the n-dimensional case follows by induction and Fubini’stheorem.
A proof of (BM) - II
Given A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ
= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
A proof of (BM) - IIGiven A,B ⊂ Rn and λ ∈ [0, 1], let
f = characteristic function of (1− λ)A + λB,
g = charact. function of A, h = charact. function of B.
Then:
f ((1− λ)x + λy)) ≥ g(x)(1−λ) h(y)λ ∀ x , y ∈ Rn.
By Prekopa-Leindler inequality
Vn((1− λ)A + λB) =
∫Rn
fdz
≥(∫
Rn
gdx
)1−λ (∫Rn
hdy
)λ= Vn(A)1−λVn(B)λ,
i.e. the multiplicative form of (BM).
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
The isoperimetric inequality
Thm. Among all subsets of Rn with given perimeter, the ballhaving such perimeter maximizes the volume.
Equivalently,
Vn(A)n−1n ≤ c(n)Hn−1(∂A)
for every set A (with sufficiently smooth boundary), where c(n) isa constant and Hn−1 is the (n − 1)-dimensional Hausdorffmeasure. Equality is attained when A is a ball.
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}
= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
(BM) ⇒ isoperimetric inequality
Let A ⊂ Rn be a bounded domain with C 1 boundary. Then
Hn−1(∂A) = limε→0+
Vn(Aε)− Vn(A)
ε,
where
Aε = {x ∈ Rn : dist(x ,A) ≤ ε}= A + εB,
andB = {x ∈ Rn : ‖x‖ ≤ 1} = unit ball.
Hence
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n
= Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A)
= c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
Proof of the isoperimetric inequality
Hn−1(∂A) = limε→0+
Vn(A + εB)− Vn(A)
ε.
By (BM), for every ε > 0
Vn(A + εB)1/n ≥ Vn(A)1/n + Vn(εB)1/n = Vn(A)1/n + εVn(B)1/n.
Vn(B)1/n ≤ limε→0+
Vn(A + εB)1/n − Vn(A)1/n
ε
=1
nVn(A)
1−nn Hn−1(∂A).
Vn(A)n−1n ≤ 1
nVn(B)1/nHn−1(∂A) = c(n)Hn−1(∂A).
When A is a ball this becomes an equality.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere;
a prototype is∫Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
The infinitesimal form of (BM)
I By the Brunn-Minkowski inequality V1/nn is a concave
functional.
I Hence the second variation (or second differential) of V1/nn
(whatever that means) must be negative semidefinite:
D2(V1/nn ) ≤ 0.
I If we restrict our attention to convex sets, this fact amountsto a class of functional inequalities of Poincare type on theunit sphere; a prototype is∫
Sn−1
φ2dHn−1 ≤ c(n)
∫Sn−1
|∇φ|2dHn−1,
∀ φ ∈ C 1(Sn−1), verifying some zero-mean condition.
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.
A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
Convex bodies
From now on we will only consider a special type of compact sets:convex bodies.A convex body is a compact convex subset of Rn. We set
Kn = {convex bodies in Rn}.
Kn is closed under addition and dilations: given K , L ∈ Kn andα, β ≥ 0,
αK + βL ∈ Kn.
The Brunn-Minkowski inequality holds in particular in Kn.
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
From sets to functions:the support function of a convex body
I The support function hK of a convex body K is defined by:
hK : Sn−1 → R , hK (u) = sup{(u, v) |v ∈ K} .
hK (u) is the distance from the origin of the hyperplanesupporting K , with outer unit normal u.
I The passage to support functions preserves the linearstructure on Kn:
hαK+βL = αhK + βhL .
for every K , L ∈ Kn α, β ≥ 0.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) ,
(hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}
= {support functions of C 2+ convex bodies}.
Convex bodies of class C 2+
A convex body is said to be of class C 2+ if:
I ∂K ∈ C 2,
I the Gauss curvature is strictly positive on ∂K .
In terms of the support function h of K :
h ∈ C 2(Sn−1) , (hij + hδij) > 0 on Sn−1
(hij = second covariant derivatives of h on Sn−1, δij =Kronecker’s symbols).
C := {h ∈ C 2(Sn−1) : (hij + hδij) > 0 on Sn−1}= {support functions of C 2
+ convex bodies}.
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
A representation formula for the volume
If K is of class C 2+ and h is its support function, then
Vn(K ) =1
n
∫Sn−1
h det(hij + hδij) dHn−1.
Now we define a functional F : C → R+ as
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
= Vn(K )1/n.
By the Brunn-Minkowski inequality,
F is concave in C.
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ)
=d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
The second variation of F.
F(h) =
[1
n
∫Sn−1
h det(hij + hδij) dHn−1]1/n
.
For every fixed h, D2F(h) is a bilinear symmetric form acting ontest functions φ ∈ C∞(Sn−1):
(D2F(h)φ, φ) =d2
ds2F(h + sφ)|s=0.
The condition(D2F(h)φ, φ) ≤ 0
turns out to be equivalent to a weighted Poincare inequality:∫Sn−1
trace(cij)φ2dHn−1 ≤
∫Sn−1
∑i ,j
cijφiφjdHn−1,
(cij) > 0, (cij) depends on h.
for every φ verifying a zero-mean condition. .
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.
(C. 2008; Saorın-Gomez, C. 2010).
A special case
If we choose h ≡ 1 (the support function of the unit ball of Rn),we obtain (cij) =identity matrix, and we recover∫
Sn−1
φ2dHn−1 ≤ 1
n − 1
∫Sn−1
|∇φ|2dHn−1,
for every φ ∈ C 1(Sn−1) s.t.∫Sn−1
φdHn−1 = 0.
This is the standard Poincare inequality (with best constant) onSn−1.(C. 2008; Saorın-Gomez, C. 2010).
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
Inequalities of Brunn-Minkowski type
Let G : Kn → R be s.t.:
I G(K ) ≥ 0 for every K ∈ Kn;
I G is α-homogeneous (α 6= 0):
G(tK ) = tα G(K ), ∀ t ≥ 0, K ∈ Kn.
—————–
We say that G verifies a Brunn-Minkowski type inequality if forevery K , L ∈ Kn, and for every λ ∈ [0, 1]:
G((1− λ)K + λL)1/α ≥ (1− λ)G(K )1/α + λG(L)1/α.
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Examples
The following functionals verify a Brunn-Minkowski type inequality.
I Volume.
I Perimeter.
I Other functionals in convex geometry (intrinsic volumes,mixed volumes, 2-dim. affine surface area...).
I Principal frequency (= first Dirichlet eigenvalue of the Laplaceoperator) (Brascamp and Lieb; Borell).
I Electrostatic capacity (Borell; Caffarelli, Jerison and Lieb).
I Many other examples coming from the world of calculus ofvariations and elliptic PDE’s (torsional rigidity, p-capacity, ...).
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
Other examples
Some well-known functional not obeying a Brunn-Minkowskiinequality.
I The diameter.
I The affine surface area in dimension n ≥ 3.
I The first Neumann eigenvalue of the Laplace operator.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?
Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
Hints
Is there some general phenomenon behind these examples?Difficult (pointless?) to say.
Maybe simpler: understand the relation between theBrunn-Minkowski inequality and other basic features, such as:
I monotonicity;
I continuity;
I rigid motion invariance;
I additivity (or valuation property):
G(K ∪ L) = G(K ) + G(L)− G(K ∩ L),
for every K , L ∈ Kn such that
K ∪ L ∈ Kn.
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.Then G is a mixed volume, and in particular is monotone.
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.
Then G is a mixed volume, and in particular is monotone.
A result in this direction
Thm. (Hug, Saorın-Gomez, C., 2012). Let G : Kn → R be:additive, rigid motion invariant, continuous, (n − 1)-homogeneous,and assume that it verifies a Brunn-Minkowski type inequality.Then G is a mixed volume, and in particular is monotone.