[14th Proceeding] 02. G.zhang

7/30/2019 [14th Proceeding] 02. G.zhang

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Proceedings of The Fourteenth International

Workshop on Diff. Geom. 14(2010) 13-30

A lecture on integral geometry

Gaoyong Zhang

Department of Mathematics, Polytechnic Institute of New York University,

6 Metrotech Center, Brooklyn, NY 11201, USA

e-mail : [email protected]

(2010 Mathematics Subject Classification : 53C65, 52A22.)

Abstract. The Steiner formula and the principal kinematic formula are two fundamen-

tal formulas in integral geometry. They are important formulas for mixed volumes andquermassintegrals of convex bodies. In this lecture, the Steiner formula is proved by a

differential geometric approach. Then the principal kinematic formula for convex bo dies

is proved by using the Steiner formula.

1 Introduction

Integral geometry originated from geometric probability. It studies random ge-ometric objects in a probability space endowed with a measure that is invariantunder a group of transformations. The geometric objects are points, lines, planes,

solids, curves, surfaces, geodesics, etc. Works of Crofton, Poincare, Sylvester, andothers set up the early stage of integral geometry. Poincare defined the basic con-cept of kinematic density by considering it as a group invariant. This led to thestudy of measures on geometric objects invariant under a group. Integral geometryhas developed into an elegant mathematical discipline by the works of Blaschke,Weil, Chern, Santalo, and others. It is closely related to convex geometry andglobal differential geometry. It is fundamental for stochastic geometry, geometrictomography and stereology. Integral geometry has found applications in many sub-jects other than mathematics, such as imaging science, material science, biologicalscience, medical science, and information science. Santalos book [6] in 1976 is amilestone of integral geometry and is still the most important book on the subject.Howards book [2] in 1993 studies integral formulas in Riemannian geometry. Thebook [4] of Klain and Rota in 1997 explains the connections of integral geometry

and convex geometry. The books [3, 5, 9] deal with various geometric probabilitiesin the Euclidean space. The book [8] by Schneider and Weil in 2008 includes recent

Key words and phrases: Kinematic formula, Steiner formula, Convex body, Mixed

volume, Quermassintegral, Curvature integral, Geometric probability, Invariant measure,

Random convex set.

13


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14 Gaoyong Zhang

developments of stochastic and integral geometry. This lecture gives a quick intro-

duction to integral geometry. The Steiner formula and the fundamental kinematicformula are proved.

2 Geometric probability

2.1 The Buffon needle problem

In 1777, the year that Gauss was born, Buffon stated and solved the followingproblem: A needle of length l is dropped at random onto a plane that is covered withparallel lines at a distance d apart. What is the probability that the needle intersectswith at least one of the lines?

The Buffon needle problem is viewed as the starting point of the subject of

geometric probability. The probability theory started much earlier in the sixteenthcentury. In early eighteenth century, Abraham de Moivre studied the normal dis-tribution.

If the length l of the needle is smaller than the distance d of parallel lines, thesolution to the Buffon needle problem is

p =2l

d.

The solution will be shown by an integral formula.

2.2 Invariant measure of sets of lines in Rn

For sets of points in Rn, the Lebesgue measure is the unique measure that isinvariant under the group G(n) of rigid motions. The Lebesgue measure of a set ofpoints in Rn is called the volume of the set. Besides sets of points, one can considersets of lines. Let Gr(1, n) be the set of all lines in Rn. This is a homogeneous space,and is called the affine projective space. The usual projective space is the set of alllines passing through the origin in Rn, which is the Grassmannian Gr(1, n). Fora line L Gr(1, n), there is a unique line Lo Gr(1, n) that is parallel to L. Letx = L Lo , where L

o is the (n 1)-subspace orthogonal to Lo. The measure

in Gr(1, n) that is invariant under the group of motions G(n) has the followingformula,

(2.2.1) dL = dx dLo,

where dLo is the invariant probability measure of the Grassmannian Gr(1, n) anddx is the Lebesgue measure in Lo.

2.3 Integral formulas of lines and planes

Let C be a compact convex set in Rn with volume V(C) and surface area S(C).Denote by P a point ofRn and by dP the Lebesgue measure. The set of points


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A lecture on integral geometry 15

that intersect with C is the set C. This gives thatCP=

dP = V(C).

The invariant measure of the set of lines that intersect with C is given by the integralformula,

(2.3.1)

CL=

dL =bn12nbn

S(C),

where bn is the volume of the unit ball in Rn. This formula can be shown as follows.

For x (C) L, let (x) be the outer unit normal of C at x. The projection ofthe surface area element dS(x) of C at x into the subspace Lo is dx

,

(2.3.2) |Lo (x)|dS(x) = dx.

Denote by u a point on the unit sphere Sn1 and by du the surface area element ofSn1. Note that a line L intersects with the boundary C at two points and dLocan be identified with the surface area element du of a half-sphere. By (2.2.1) and(2.3.2),

CL=

dL =

CL=

dx dLo

=1

2

Gr(1,n)

xC

|Lo (x)|dS(x) dLo

=1

2

xC

Gr(1,n)

|Lo (x)|dLo

dS(x)

= 12

xC

1

2nbn

Sn1

|u (x)|du

dS(x)

=1

2

xC

bn1nbn

dS(x)

=bn12nbn

S(C).

As an application, the integral formula (2.3.1) gives the solution to the Buffonneedle problem. Let C be a needle of length l. Then S(C) = 2l. One can letthe needle be fixed and let parallel lines be placed at random. The measure of allpossible parallel lines is d. Thus, the probability that the needle meets with a lineis

p =CL=

dL

d = 2ld .

The integral formula (2.3.1) for lines can be extended to an integral formula fork-planes. Let Lk be a k-dimensional plane in R

n. There is the following integralformula,

(2.3.3)

CLk=

dLk = bn,k Wk(C),


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16 Gaoyong Zhang

where bn,k is a constant that depends only on n, k, and Wk(C) is the k-th quermass-

integral of C. When C is smooth, quermassintegrals are the same as the integralsof elementary symmetric curvature functions. For example, nW2(C) is the integralof mean curvature.

2.4 Probability of detecting an object

Let C0, C1 be two convex compact sets in Rn such that C0 C1. Assume that

a random compact convex set D intersects with C1. What is the probability thatD meets with C0?

The answer is given by the formula

p(C0 D = |C1 D = ) =

nk=0

(nk

)Wk(C0)Wnk(D)

nk=0 (nk)Wk(C1)Wnk(D)

,

which follows from the fundamental kinematic formula in integral geometry,

gG(n)

(C gD) dg = b1n

nk=0

n

k

Wk(C)Wnk(D).

This formula is due to Blaschke, Santalo and Chern. We will give a new proof.

3 Convex sets

3.1 Convex sets

A set C Rn is convex if for any two points x, y C the line segment [x, y]joining them is contained in C, that is,

(1 )x + y C, 0 1.

The intersection of convex sets is convex.For x1, x2, . . . , xk R

n, any point x = 1x1 + 2x2 + + kxk, 1, . . . , k 0,1 + 2 + + k = 1, is called a convex combination of x1, x2, . . . , xk.

The convex hull of a set A, denoted by convA is the set of all convex combi-nations of points of A. It is also the intersection of all convex sets that containA.

A polytope is the convex hull of a finite set in Rn. A compact convex set withnon-empty interior is called a convex body.

3.2 The support function of a compact convex set

If C is a compact convex set in Rn, its support function h(C, ) : Rn R isdefined by

(3.2.1) h(C, x) = max{x y : y C}.


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A support function is homogeneous of degree 1 and is convex. Thus a support

function is sublinear,h(C,x) = h(C, x), 0,

h(C, x + y) h(C, x) + h(C, y).

The support function h(C, x) of C is also denoted by hC(x).

3.3 The Minkowski addition

IfC, D are compact convex sets in Rn and R, the Minkowski additionC+Dis the vector addition,

C+ D = {x + y : x C, y D},

and the scalar multiplication C is

C = {x : x C}.

Proposition 3.3.1. If C, D are compact convex sets inRn, then

h(C+ D, ) = h(C, ) + h(D, ),(3.3.1)

C+ D =xC

(x + D).(3.3.2)

Proof. By the definition of Minkowski addition, points in C+ D can be written as

y + z, y C, z D. Then by the definition of support function, for x Rn,

h(C+ D, x) = max{x (y + z) : y C, z D}

= max{x y : y C} + max{x z : z D}

= h(C, x) + h(D, x).

C + D =

xC,yD

{x + y} =xC

(x + D).

Outer parallel sets. For a compact convex set C, its outer parallel set of distance 0 is the Minkowski sum C+ B, where B is the unit ball. The support functionof the outer parallel set is

h(C+ B, ) = h(C, ) + .


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18 Gaoyong Zhang

4 Hypersurfaces

4.1 Fundamental equations

Let M be a hypersurface in Rn of class C2. For x M, let TxM be the tangentspace of M at x. Choose an orthonormal frame {e1, . . . , en} at x in R

n so that{e1, . . . , en1} is a basis of the tangent space TxM. Then en is orthogonal to TxM.The differential dx is a vector in TxM, and thus can be expressed by the basis{e1, . . . , en1},

dx =n1i=1

iei,

where i are certain differential 1-forms.Taking the differential of en en = 1 gives en den = 0. Thus, den is orthogonal

to en, and can be expressed by the basis {e1, . . . , en1},

den =

n1i=1

niei,

where ni are certain differential 1-forms.Express the differential dei by {e1, . . . , en},

dei =n1j=1

ijej + inen,

where ij and in are certain differential 1-forms.Differentiating ei ej = ij and en ei = 0 give

dei ej + ei dej = 0,

dei en + ei den = 0.

Then

ij + ji = 0,

ni + in = 0.

The Fundamental equations of a hypersurface are the following,

dx =n1

i=1

iei,(4.1.1)

dei =

n1j=1

ijej + inen, ij + ji = 0,(4.1.2)

den = n1i=1

inei.(4.1.3)


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4.2 Structure equations

The 1-forms {i}, {ij}, and {in} satisfy certain equations, called structureequations. The equation d2x = 0 and the fundamental equations give structureequations,

(4.2.1) di =n1j=1

ij j ,

and

(4.2.2)

n1i=1

i in = 0.

The last equation implies that

(4.2.3) in =n1j=1

bijj , bij = bji .

Similarly, the equations d2ei = 0 give structure equations,

dij =n1k=1

ik kj in jn ,(4.2.4)

din =

n1j=1

ij jn ,(4.2.5)

where (4.2.4) is called the Gauss equationand (4.2.5) is called the Codazzi equation.Plugging (4.2.3) into (4.2.5) gives

n1j=1

(dbij

n1k=1

bikjk n1k=1

bkjik) j = 0.

Therefore, the Codazzi equation becomes

(4.2.6) dbij n1k=1

bikjk n1k=1

bkjik =

n1k=1

bijkk, bijk = bikj .

4.3 Fundamental forms

The first fundamental form of the hypersurface M is defined as

(4.3.1) I = dx dx =n1i=1

2i .

The surface area element of M is defined as

(4.3.2) dS = 1 n1.


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It is easily verified that both the first fundamental form and the surface area element

are independent of the choice of orthonormal frames. The first fundamental formis the metric on the hypersurface induced from the metric ofRn.The second fundamental form of the hypersurface M is defined as

(4.3.3) II = dx den =n1i=1

iin,

and the third fundamental form of the hypersurface M is defined as

(4.3.4) III = den den =n1i=1

2in.

The second and the third fundamental forms are also independent of the choice oforthonormal frames. But the second fundamental form changes its sign when thenormal changes its sign.

4.4 Curvatures

For a tangent vector v TxM, the normal curvature n(v) of the hypersurfaceM at point x in direction v is defined by

(4.4.1) n(v) =II(v)

I(v).

The normal curvature is the curvature of the normal section curve at x that is the

intersection of M with the two dimensional plane spanned by the tangent vector vand the normal vector en.

The fundamental equation (4.1.3) can be viewed as a linear transformation inthe tangent space TxM. Let

(4.4.2) W(v) = den(v) =n1i=1

in(v)ei, v TxM.

The linear transformation W is called the Weingarten transformation. By (4.2.3),we have

(4.4.3) W(ei) =

n1

j=1

bijej.

Since bij = bji , the Weingarten transformation is self-adjoint. The eigenvalues1, . . . , n1 of the Weingarten transformation W are called the principal curvaturesof the hypersurface M. The principal curvatures are solutions of the polynomialequation,

det(ij bij) = 0.


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Write this polynomial as

det(ij bij) =n1k=0

(1)k

n 1

k

Hk

n1k.

The function Hk on M is the kth normalized elementary symmetric function of theprincipal curvatures,

(4.4.4) Hk =

n 1

k

1 1i1


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4.6 The support function of a hypersurface

Let M be a C2

hypersurface in Rn

. Its support function is defined as

(4.6.1) hM(x) = x en = x (x), x M.

Suppose hM is C2 and the third fundamental form III is not degenerated. Then

the Hessian matrix of hM with respect to the third fundamental form III can becomputed. Write the differential dh as

dhM =n1i=1

hiin.

Differentiating (4.6.1) and using (4.1.3) give

hi = x ei.

Then

dhi =n1j=1

hjij + i hin.

That the third fundamental form III is not degenerated is equivalent to that theGauss curvature is not zero. Thus the matrix (bij) is invertible. Denote by (cij)

the inverse of (bij). Then i =n1

j=1 cijjn . Therefore, the Hessian matrix of h is(hij) = (cij hij), and thus

(4.6.2) (cij) = (hij + hij).

where the Hessian matrix (hij) is defined by

dhi =n1j=1

hjij +n1j=1

hijjn .

It follows that the reciprocal Gauss curvature has the following formula,

(4.6.3)1

Hn1= det(hij + hij).

The sum of the reciprocal principal curvatures has the formula,

(4.6.4)1

1+ . . . +

1

n1= h + (n 1)h,

where is the Laplacian of the third fundamental form.Let ri =

1i

, i = 1, . . . , n 1, called the radii of principal curvature. They arethe eigenvalues of the matrix (hij + hij). Let

(4.6.5) Fk =

n 1

k

1 1i1


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The functions Hk and Fk satisfy

Fk =Hn1k

Hn1,(4.6.6)

Hk =Fn1k

Fn1.(4.6.7)

Sometimes it is convenient to use Fk as a function on the unit sphere Sn1, thatis to use Fk(

1()), called the kth curvature function of K. Without confusing, letFk(u) = Fk(

1(u)) for u Sn1.Define the kth curvature integral Mk(K), k = 0, 1, . . . , n 1, of a convex body

K by

(4.6.8) Mk(K) = xK Hk(x)dS(x),which can also be written

(4.6.9) Mk(K) =

Sn1

Fn1k(u)du.

Define the kth quermassintegral Wk(K), k = 0, 1, . . . , n 1, of a convex bodyK by

(4.6.10) Wk(K) =1

n

Sn1

h(K, u)Fn1k(u)du.

Denote the volume of a convex body C by V(C), and denote the surface areaof C by S(C). Then

S(C) = nW1(C),(4.6.11)

V(C) = W0(C).(4.6.12)

Define Wn(K) = V(B), the volume of the unit ball.

5 Steiners formula

5.1 Volume formulas

The volume of a compact set in Rn is its Lebesgue measure. Denote the volumefunctional by V().

Lemma 5.1.1. If M is a compact domain in Rn whose boundary is a closed hy-

persurface, then

(5.1.1) V(M) =1

n

M

x (x) dS(x).


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24 Gaoyong Zhang

Proof. The formula follows immediately from the divergence formula,xM

x (x)dS(x) =

xM

ndx.

The volume of a convex body has an integral formula in terms of the supportfunction.

Theorem 5.1.2. Let C be a convex body of C2 boundary and positive curvature.

If h is the support function of C defined on the unit sphere Sn1 and (hij) is the

Hessian matrix of h onSn1, then

(5.1.2) V(C) =1

n

Sn1

h det(hij + hij) du.

Proof. Let : C Sn1 be the Gauss map. Write u = (x). Then the surface

area element dS(x) of C at x and the surface area element du of Sn1 at u are

related by

dS(x) =1

Hn1du,

where Hn1 is the Gauss curvature of C at x. By the equation (4.6.3),

(5.1.3) dS(x) = det(hij + hij) du.

The support function h of C can be written as

(5.1.4) h(u) = 1(u) u = x (x).

Thus the volume formula (5.1.2) follows from the volume formula (5.1.1), equations

(5.1.3) and (5.1.4).

5.2 Variation of volume

For convex bodies C, D, denote by Ct the Minkowski sum C+ tD, t > 0.

Lemma 5.2.1. Let C, D be C2 convex bodies of positive curvature. Then

(5.2.1) limt0+

V(Ct) V(C)

t=

xC

h(D, C(x)) dS(x),

where C(x) is the outer unit normal of C at x.


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Proof. When t is small, the boundary points of Ct are xt = x + t 1D (C(x)),

x C, where 1D is the inverse Gauss map of D. The outer unit normal of Ct

at xt is the same as C(x). The volume element of Ct is dSt(xt) = dS(x) + O(t).

The normal distance of x and xt is (xt x) C(x) = t h(L, C(x)). Thus, the

volume of the set Ct \C is tC

h(D, C(x)) dS(x) + O(t2). It follows that

V(Ct) V(C) = t

C

h(L, C(x)) dS(x) + O(t2).

This gives the formula (5.2.1).

5.3 Mixed curvature functions and mixed volumes

Let C, D be convex bodies whose boundaries are ofC2 and of positive curvature.Let

cij = hij(C, ) + h(C, )ij ,

dij = hij(D, ) + h(D, )ij .

As noted before, (cij) and (dij) are inverse matrices of the second fundamental formsof C and D for orthonormal frames. The Minkowski sum C + tD has supportfunction h(C, ) + th(D, ). Therefore, the inverse matrix of the second fundamentalform of the closed hypersurface (C+ tD) for an orthonormal frame is (cij + tdij).Consider the expansion,

(5.3.1) det(cij + tdij) =n1k=0

n 1

k

Fn1k(C, D; )t

k.

The coefficients Fn1k(C, D; ), k = 0, 1, 2, . . . , n1, are called the mixed curvaturefunctions of C and D. When D is the unit ball B, Fn1k(C, B) are the curvaturefunctions of C.

Define the mixed curvature integral Mk(C, D) by

(5.3.2) Mk(C, D) =

Sn1

h(D, u)Fn1k(C, D; u) du.

When D is the unit ball B, the mixed curvature integral Mk(C, B) becomes thecurvature integral of M

k(C).

Define the mixed quermassintegralor mixed volume Vk(C, D) by

(5.3.3) Vk(C, D) =1

n

Sn1

h(C, u)Fn1k(C, D; u) du.

When D is the unit ball B, the mixed quermassintegral Vk(C, B) becomes thequermassintegral Wk(C).


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26 Gaoyong Zhang

5.4 Steiners formula

Let C, D be convex bodies in Rn

, and

Ct = C+ tD, t 0.

Suppose that C,D are of class C2 and of positive curvature. Then the reciprocalGauss curvature of Ct is det(cij + tdij). Therefore, by (5.1.2), (5.3.1), (5.3.2) and(5.3.3),

V(Ct) =1

n

Sn1

(h(C, u) + th(D, u)) det(cij + tdij)du

=1

n

Sn1

(h(C, u) + th(D, u))

n1

k=0

n 1

k

Fn1k(C, D; u)t

kdu

=n1k=0

n 1

k

Vk(C, D)t

k +1

n

n1k=0

n 1

k

Mk(C, D)t

k+1

= V(C) +

n2k=0

n 1

k + 1

Vk+1(C, D)t

k+1 +1

n

n1k=0

n 1

k

Mk(C, D)t

k+1.

On the other hand, by (5.2.1), (5.3.1) and (5.3.2) there is

V(Ct) = V(C) +

t0

dV(C)

dd

= V(C) + t

0C

h(D, C(x))dS(x)d

= V(C) +

t0

Sn1

h(D, u)det(cij + dij)dud

= V(C) +

t0

Sn1

h(D, u)n1k=0

n 1

k

Fn1k(C, D; u)

kdud

= V(C) +n1k=0

n 1

k

1

k + 1Mk(C, D)t

k+1.

Comparing coefficients of both polynomials gives

n 1k + 1

Vk+1(C, D) +

1

nn 1

k

Mk(C, D) =n 1

k 1

k + 1 Mk(C, D).

This gives the following Minkowskis formulas,

(5.4.1) nVk+1(C, D) = Mk(C, D), k = 0, 1, 2, . . . , n 2.

This and the above polynomials give the following Steiner formulaof mixed volumes.


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Theorem 5.4.1. If C, D are convex bodies inRn, then for t 0,

(5.4.2) V(C+ tD) =n

k=0

n

k

Vk(C, D) t

k.

A consequence of the formula (5.4.2) is the Steiner formula of quermassintegralsfor outer parallel bodies.

Corollary 5.4.2. If C is convex body and B is the unit ball inRn, then for t 0,

(5.4.3) V(C+ tB) =

nk=0

n

k

Wk(C) t

k.

6 The fundamental kinematic formula

6.1 Elementary symmetric functions

Let (aij) be an (n 1) (n 1) matrix whose eigenvalues are real numbers1, 2, . . . , n1. Define the k-th elementary symmetric function,

(6.1.1) k =

i1


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28 Gaoyong Zhang

Lemma 6.2.1. If C, D are convex bodies inRn, then

(6.2.1)SO(n)

Vk(C,D) d = b1n Wk(C)Wnk(D).

Proof. Let (cij) and (dij) be the inverse matrices of the second fundamental forms

of C and (D) for orthonormal frames. Let (aij) = (dij)1(cij). By (5.3.1),

det(dij)det(aij + tij) =

n1k=0

n 1

k

Fn1k(C,D; )t

k.

By (6.1.2),

k =

n 1

k

Fk(C,D; )/ det(dij).

Choose the orthonormal frame so that (dij)1

= diag{1, . . . , n1}, where i arethe principal curvatures of (D). Then (aij) = (icij). By (6.1.3),

k =1

k!

i1,...,ik;j1,...,jkci1j1 cikjki1 ik .

Diagonalizing the matrix (cij) = TtT, where = diag{1, . . . , n1} and T =

(ij) is orthogonal. Note that 1, . . . , n1 are the radii of curvature of C. Then

k =1

k!

i1,...,ik;j1,...,jkl1i1l1j1 lkiklkjkl1 lki1 ik ,

where the indices i1, . . . , ik are distinct, (j1, . . . , jk) is a permutation of (i1, . . . , ik).

The indices l1, . . . , lk can be assumed distinct. For example, ifl1 = l2, the two termscorrespond to (j1, j2, . . . , jk) and (j2, j1, . . . , jk) have opposite signs and cancel out.

By (5.3.3),SO(n)

Vn1k(C,D) d =1

n

n 1

k

1 SO(n)

Sn1

h(C, u)det(dij)kdud.

Note that det(dij) = (1(D,u) n1(D,u))1. Let u and the indices l1, . . . , lk

be fixed and compute the following integral,

A =

SO(n)

i1,...,ik;j1,...,jkl1i1l1j1 lkiklkjki1(D,u) ik(D,u)

1(D,u) n1(D,u)d

This integral is a constant independent of u. Integrate first over the rotation group

SO(n 1) in the subspace u orthogonal to u. Then for fixed distinct indices

i1, . . . , ik, and distinct indices l1, . . . , lk, the integralSO(n1)

j1,...,jk

i1,...,ik;j1,...,jkl1i1l1j1 lkiklkjkd


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is a constant depending only upon n, k because the orthogonal matrix (ij) runs

over all rotations in SO(n 1). Therefore,

A = c(n, k)Mk(D)

for certain constant c(n, k). It follows that

SO(n)

Vn1k(C,D) d =1

n

n 1

k

1 Sn1

h(C, u)Fk(C, u)c(n, k)Mk(D) du

= c(n, k)n 1

k 1

Wn1k(C)Mk(D)

= nc(n, k)

n 1

k

1Wn1k(C)Wk+1(D).

The exact constant can be obtained by letting D be the unit ball.

6.3 The fundamental kinematic formula

Denote by G(n) the group of rigid motions in Rn. Every element g G(n)is determined by a translation x Rn and a rotation SO(n). The invariantmeasure of G(n) is defined as dg = dxd where dx is the Lebesgue measure ofRn

and d is the invariant probability measure of SO(n).

Theorem 6.3.1. If C, D are convex bodies inRn, then

(6.3.1)

G(n)

(C gD) dg = b1n

nk=0

n

k

Wk(C)Wnk(D).

Proof. Note that (C (x + D)) = 0, or 1, and (C (x + D)) = 1 if and only if

C (x + D) = , that is, x C+ (D). Thus,

Rn

(C (x + D)) dx = V(C+ (D)).

By the Steiner formula of mixed volumes,

Rn

(C (x + D)) dx =n

k=0

n

k

Vk(C,D).


18/18

30 Gaoyong Zhang

By Lemma 6.2.1,G(n)

(C gD) dg =

SO(n)

Rn

(C (x + D) dxd

=

SO(n)

nk=0

n

k

Vk(C,D)d

= b1n

nk=0

n

k

Wk(C)Wnk(D).

Acknowledgement. This lecture was given in a workshop organized by ProfessorYoung Jin Suh in December, 2009, at Kyungpook National University, Daegu, Ko-rea. The author would like to thank Professor Suh for the invitation and to thankhim and his students for their hospitality.

References

[1] S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando, 1984.

[2] R. Howard The Kinematic Formula in Riemannian Geometry, Mem. Amer.Math. Soc., No. 509, Vol. 106, 1993.

[3] M. Kendall and P. Moran, Geometric Probability, Charles Griffin, London,1963.

[4] D. Klain and G.C. Rota, Introduction to Geometric Probability, CambridgeUniversity Press, Cambridge, 1997.

[5] Ren De-lin, Topics in Integral Geometry, World Scientific, 1994.

[6] L. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley,Reading, MA, 1976.

[7] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Uni-versity Press, Cambridge, 1993.

[8] R. Schneider and W. Weil, Stochastic and Integral Geometry, Springer, 2008.

[9] H. Solomon, Geometric Probability, Soc. Industr. Appl. Math., Philadelphia,1978.

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