Embed Size (px)
Citation preview
Metric Thickenings, Orbitopes, and
Borsuk–Ulam Theorems
GTDAML 2019
Johnathan Bush†
Joint work with Henry Adams† and Florian Frick‡
The Ohio State University, June 2nd, 2019
†Department of Mathematics, Colorado State University‡Department of Mathematical Sciences, Carnegie Mellon University
Metric Vietoris–Rips Thickenings
Definition (Adamaszek, Adams, Frick [1])
For a metric space X and r ≥ 0, the Vietoris–Rips thickeningVRm(X; r) is the set
VRm(X; r) =
{k∑
i=0
λixi
∣∣∣∣ k ∈ N, xi ∈ X, and diam({x0, . . . , xk}) ≤ r
}
equipped with the 1-Wasserstein metric.
1/3 1/3 1/61/6
1/31/3 1/3
2
VRm(X; r) at small scale parameters
Theorem (Adams, Mirth [2])
Let X ⊆ Rn with positive reach. Then,
VRm(X; r) ' X
for r sufficiently small.
3
VRm(X; r) at large scale parameters
Conjecture
Let S1 be the circle of unit circumference. Then, for k ∈ N,
VRm(S1; r) ' S2k−1 ifk − 1
2k − 1≤ r < k
2k + 1.
0
S1 S3 S5 S7
13
25
3749
r =1
3 r =2
5r =
3
7
4
VRm(X; r) at large scale parameters
Conjecture
Let S1 be the circle of unit circumference. Then, for k ∈ N,
VRm(S1; r) ' S2k−1 ifk − 1
2k − 1≤ r < k
2k + 1.
�������
��
��
��
��
��
Torus.nb 7
�������
��
��
��
��
��
6 Torus.nb
�������
��
��
��
��
��
Torus.nb 5
�������
��
��
��
��
��
4 Torus.nb
�������
��
��
��
��
��
Torus.nb 3
�������
��
��
��
��
��
2 Torus.nb
r =1
3 r =2
5r =
3
7
4
VRm(X; r) at large scale parameters
Conjecture
VRm(S1; r) ' S2k−1 ifk − 1
2k − 1≤ r < k
2k + 1.
Desired proof.
For k−12k−1 ≤ r <
k2k+1 , there exist homotopy equivalences
p ◦ SM2k and ι in the following diagram:
VRm(S1; r)SM2k−−−→ R2k \ {~0} p−→ ∂B2k
ι−→ VRm(S1; r),
where p denotes the radial projection, ι denotes the inclusion,
and ∂B2k ∼= S2k−1.
5
Barvinok–Novik Orbitopes
Fix k ≥ 1 and define the symmetric moment curve
SM2k : R/2πZ→ R2k
θ 7→ (cos(θ), sin(θ), cos(3θ), sin(3θ), . . . , cos((2k − 1)θ), sin((2k − 1)θ)).
Define the k-th Barvinok–Novik orbitope by
B2k = conv(SM2k(S1)).
6
Barvinok–Novik Orbitopes
Fix k ≥ 1 and define the symmetric moment curve
SM2k : R/2πZ→ R2k
θ 7→ (cos(θ), sin(θ), cos(3θ), sin(3θ), . . . , cos((2k − 1)θ), sin((2k − 1)θ)).
Define the k-th Barvinok–Novik orbitope by
B2k = conv(SM2k(S1)).
6
Barvinok–Novik Orbitopes
Theorem (Barvinok, Novik [3])
The proper faces of B4 are
• the 0-dimensional faces, SM4(t) for t ∈ S1,
• the 1-dimensional faces, conv(SM4({t1, t2})) where t1 6= t2
are the edges of an arc of S1 of length ≤ 13 ,
• the 2-dimensional faces, conv(SM4({t, t+ 13 , t+ 2
3})) for
t ∈ S1.
The precise facial structure of B2k is
unknown for k > 2.
Known to be simplicial and locally k-
neighborly ([5], [3]).
7
Barvinok–Novik Orbitopes
Theorem (Barvinok, Novik [3])
The proper faces of B4 are
• the 0-dimensional faces, SM4(t) for t ∈ S1,
• the 1-dimensional faces, conv(SM4({t1, t2})) where t1 6= t2
are the edges of an arc of S1 of length ≤ 13 ,
• the 2-dimensional faces, conv(SM4({t, t+ 13 , t+ 2
3})) for
t ∈ S1.
The precise facial structure of B2k is
unknown for k > 2.
Known to be simplicial and locally k-
neighborly ([5], [3]).
7
VRm(X; r) at large scale parameters
Conjecture
VRm(S1; r) ' S2k−1 ifk − 1
2k − 1≤ r < k
2k + 1.
Desired proof.
For k−12k−1 ≤ r <
k2k+1 , there exist homotopy equivalences
p ◦ SM2k and ι in the following diagram:
VRm(S1; r)SM2k−−−→ R2k \ {~0} p−→ ∂B2k
ι−→ VRm(S1; r),
where p denotes the radial projection, ι denotes the inclusion,
and ∂B2k ∼= S2k−1.
8
VRm(X; r) at large scale parameters
So far:
Theorem (Adams, B., Frick)
VRm(S1; 1/3) ' S3.
Proof.
There exist homotopy equivalences p ◦ SM4 and ι in the
following diagram:
VRm(S1; 1/3)SM4−−−→ R4 \ {~0} p−→ ∂B4
ι−→ VRm(S1; 1/3)
where p denotes the radial projection, ι denotes the inclusion,
and ∂B4 ∼= S3.
9
Borsuk–Ulam Theorems
Theorem (Borsuk–Ulam)
Given a continuous function f : Sn → Rn, there exists x ∈ Sn
such that f(x) = f(−x).
Equivalently, given a continuous and odd function f : Sn → Rn,
there exists x ∈ Sn such that f(x) = ~0.
10
Borsuk–Ulam Theorems
Theorem (Gromov [4])
Given a continuous function f : Sn → Rk with k ≤ n, there
exists y ∈ Rk such that the n-spherical volume of the ε-tubular
neighborhood of f−1(y), denoted by f−1(y) + ε, satisfies
Voln(f−1(y) + ε) ≥ Voln(f−1(Sn−k + ε))
for every ε > 0.
11
Borsuk–Ulam Theorems
Theorem (Adams, B., Frick)
If f : S1 → R2k+1 is continuous, there exists a subset
{x1, . . . , xm} ⊆ S1 of diameter at most k2k+1 and with
m ≤ 2k + 1 such that∑m
i=1 λif(xi) =∑m
i=1 λif(−xi), for some
choice of convex coefficients λi.
Equivalently, if f : S1 → R2k+1 is continuous and odd, then
there exists a subset X ⊆ S1 of diameter at most k2k+1 and size
|X| ≤ 2k + 1 such that ~0 ∈ conv(f(X)).
This result is sharp: f = SM2k : S1 →R2k ⊂ R2k+1 is an odd map such that~0 /∈ conv(f(X)) if diam(X) < k
2k+1 .
12
Borsuk–Ulam Theorems
Theorem (Adams, B., Frick)
If f : S1 → R2k+1 is continuous, there exists a subset
{x1, . . . , xm} ⊆ S1 of diameter at most k2k+1 and with
m ≤ 2k + 1 such that∑m
i=1 λif(xi) =∑m
i=1 λif(−xi), for some
choice of convex coefficients λi.
Equivalently, if f : S1 → R2k+1 is continuous and odd, there
exists a subset X ⊆ S1 of diameter at most k2k+1 and size
|X| ≤ 2k + 1 such that ~0 ∈ conv(f(X)).
Proof.
The induced map f : VRm(S1; k2k+1)→ R2k+1 is odd with
domain VRm(S1; k2k+1) ' S2k+1. By Borsuk–Ulam, this map
has a zero, giving a subset X of diameter at most k2k+1 with
conv(f(X)) containing the origin.
13
Future Work
• Currently: partial results for spheres, maps Sn → Rn+2.
• Recent idea: full Borsuk–Ulam type results (i.e., tight
bounds) for odd dimensional spheres by taking n-fold joins
of S1.
• A better understanding of metric thickenings of spheres at
large scales. (Cech thickenings, different orbitopes, etc.)
14
Future Work
• Currently: partial results for spheres, maps Sn → Rn+2.
• Recent idea: full Borsuk–Ulam type results (i.e., tight
bounds) for odd dimensional spheres by taking n-fold joins
of S1.
• A better understanding of metric thickenings of spheres at
large scales. (Cech thickenings, different orbitopes, etc.)
14
Future Work
• Currently: partial results for spheres, maps Sn → Rn+2.
• Recent idea: full Borsuk–Ulam type results (i.e., tight
bounds) for odd dimensional spheres by taking n-fold joins
of S1.
• A better understanding of metric thickenings of spheres at
large scales. (Cech thickenings, different orbitopes, etc.)
14
References
[1] M. Adamaszek, H. Adams, and F. Frick, Metric reconstruction via optimal transport, To appear
in the SIAM Journal on Applied Algebra and Geometry, (2018).
[2] H. Adams and J. Mirth, Metric thickenings of euclidean submanifolds, Topology and its
Applications, 254 (2019), pp. 69–84.
[3] A. Barvinok and I. Novik, A centrally symmetric version of the cyclic polytope, Discrete &
Computational Geometry, 39 (2008), pp. 76–99.
[4] M. Gromov, Isoperimetry of waists and concentration of maps, Geometric and functional
analysis, 13 (2003), pp. 178–215.
[5] R. Sinn, Algebraic boundaries of SO(2)-orbitopes, Discrete & Computational Geometry, 50
(2013), pp. 219–235.
Thank you!
15
