On geometric measures and convexity
Jiazu Zhou, Chunna Zeng and Guangxian Zhu
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China, China
e-mail: [email protected]
(2000 Mathematics Subject Classification : 52A22, 60D05.)
Abstract. We begin with the geometric measure for geometric objects in the Euclidean space Rn. As one expected, these geometric measures are the mean curvature integrals of the smooth boundaries of domains. These geometric measures are also equivalent to the Minkowski quermassintegrals when we consider the convex bodies. The kinematic formulas, based on the study of geometric measures on geometric objects, lead to more geometric inequalities.
1 Introduction
Integral geometry, referred to as geometric probability in the past and origi- nated with the Buffon needle problem in 1770s, begins with the study of invariant measures on geometric objects, such as the set of lines in the Euclidean plane and the set of linear spaces in the Euclidean spaceRn, and leads to integral identities and inequalities that have remarkable applications to probability and other math- ematical branches (see [11], [12], [15]).
A subset K in the Euclidean space Rn is convex if any two points x, y of K are endpoints of the line segment lying inside K. The kinematic formulas, based on the study of geometric measures on geometric objects, such as the set of lines in the Euclidean plane R2, convex sets, simplex and the sets of linear spaces in the Euclidean space Rn, lead to more geometric inequalities. We also introduce the Hadwiger’s fundamental theorem of invariant measure. Finally we outline the kinematic formulas and its applications to geometric inequalities.
Definition 1. LetFbe a family of subsets of a setSin then-dimensional Euclidean spaceRn, which is closed under unions and intersections, and contains the empty Key words and phrases:Geometric probability, geometric measure, invariant measure, Grassmanian.
*) Supported in part by Chinese NSF (grant number: 10671159) and Southwest University.
33
set. A geometric measure µ is a real valued function (not necessarily positive) defined on Fand satisfies,
1. µ(∅) = 0;
2. IfA, B∈Fare two measurable sets, then
µ(A∪B) =µ(A) +µ(B)−µ(A∩B);
3. For the group of rigid motionsISO(Rn) inRn, µ(A) =µ(gA); for g∈ISO(Rn).
4. For a parallelotope P(x1,· · · , xn) with each orthogonal side parallel to the coordinate axis of Rn,
µ(P) =x1·x2· · · · ·xn
determines the solid volume ofP.
A measure is characterized by the first two axioms in Definition 1. The axiom (3) is required for the measureµbeing invariant. That is, the measure does not depend on the position of the set in space. Note that (4) is intuitively the characterized volume and is just the normalization ofµhence it is not unique.
Geometric Measure. For Definition 1, if we keep the first 3 axioms but tamper the forth axiom, the normalization axiom, what will happen?
Let us take the elementary symmetric functions of (the following polynomials in)nvariables:
e1(x1, x2,· · · , xn) =x1+x2+· · ·+xn; e2(x1, x2,· · ·, xn) =x1x2+x1x3+· · ·+xn−1xn;
· · ·
en−1(x1, x2,· · ·, xn) =x2x3· · ·xn+· · ·+x1x2· · ·xn−1;
en(x1, x2,· · ·, xn) =x1x2· · ·xn.
Lettingµk=ek(x1, x2,· · ·, xn) (k= 1,2,· · ·, n) will lead tondifferent invari- ant measures. The measure in Definition 1 normalized by the axiom (4) is justµn. Each of thenelementary symmetric functions ofnvariables leads to the definition of a new invariant measure which is a different generalization of the volume. These nmeasures are called the intrinsic measures.
The intrinsic measures are independent of each other, except for certain inequal- ities they satisfy. These inequalities generalized the classical isoperimetric inequal- ity that relates area to volume. Mathematicians are presently studying these as yet unknown geometric inequalities among the intrinsic volumes. We know very little about the intrinsic volumes and inequalities among them.
One may ask if there are other invariant measures except for these µk (k = 1,· · · , n). The answer to this question is yes. We have another very important invariant measure.
If we add another invariant measure:
µ0(C) = 1, C is a non-empty convex set,
andµ(∅) = 0, then we have the fundamental theorem of invariant geometric measure (see [11]).
Theorem 1. (Hadwiger-Klain) The n+ 1 intrinsic volume µ0, µ1,· · · , µn are a basis of the space of all continuous invariant measures defined on all finite unions of compact sets.
One would like to identify these n+ 1 intrinsic volume with Minkowski quer- massintegrales.
2 Minkowski quermassintegrale
For anmdimensional submanifoldM inRn, if we pick any pair of independent tangent vectorsuandv inTp(M), the tangent space atp∈M, then for every unit vector w = λu+µv, there is a unique geodesic in M starting at pwith tangent vectorw. The set of all such geodesics, asw describes the unit circle in the plane spanned by u and v, sweep a surface whose Gauss curvature at p is the section curvatureKΠ=K[u, v] of the plane Π spanned byuandv.
Lete1,· · ·, embe an orthonormal basis of the tangent spaceTp(M) of M atp.
Then the quantity
S= 2 X
1≤i<j≤m
K[ei, ej]
is independent of the choice of basis, and is called the scalar curvature ofM atp.
For a hypersurface Σ in Rn, we may choose e1,· · ·, en−1 to be the principal curvature directions at p. Then the scalar curvature S of Σ can be expressed in terms of the principal curvaturesκ1,· · · , κn−1 by
S= 2 X
1≤i<j≤n−1
κiκj.
One considers the Gauss map
G:p7→N(p) whose differential
dGp:x0(t)7→N0(t) wherex(0) =p, satisfies Rodrigues’ equations
dGp(ei) =−κiei, i= 1,2,· · · , n−1.
Therefore we have the mean curvature H = 1
n−1(κ1+· · ·+κn−1) =− 1
n−1trace(dGp) and the Gauss-Kronecker curvature
K=κ1· · ·κn−1= (−1)n−1det(dGp).
Thejth-order mean curvature is thejth-order elementary symmetric functions of the principal curvatures. We denote byHj thejth-order mean curvature, nor- malized such that
n−1Y
j=1
(1 +tκj) =
n−1X
j=0
µ n−1 j
¶ Hjtj. That is
Hj=
µ n−1 j
¶−1
{κi1,· · ·, κij}; j= 1,· · ·, n−1
where{κi1,· · · , κij}is thejth elementary symmetric function of the principal cur- vatures. Thus, H1 = H, the mean curvature, and Hn−1 is the Gauss-Kronecker curvature.
Forn= 3, that is all about it, but in higher dimensions, there are intermediate Hj. Among them,H2 plays an important role in differential geometry. The jth- order integral of mean curvatureMj(Σ) is defined by
Mj(Σ) = Z
Σ
HjdA j= 1,· · · , n−1,
where dA is the volume element of Σ. We let M0(Σ) = A, the area of Σ for completeness.
A support hyperplane of a convex setK(or a support hyperplane of the convex surface ∂K) is a hyperplane that contains points of K but does not seperate any two points ofK. LetKbe a convex set and letO be a fixed point inRn. Consider all the (n−r)-planeLn−r[O] throughO and letKn−r0 be the orthogonal projection ofKintoLn−r[O]. That is,Kn−r0 denotes the convex set of all intersection points of Ln−r[O] with ther-plane perpendicular toLn−r[O] through each point ofK. Then the r-th Minkowski quermassintegrale Wr(K), or mean cross-sectional measure, introduced by Minkowski, is defined by the normalizedE(V(Kn−r0 )):
E(V(Kn−r0 )) = Ir(K) m(Gn−r,r),
where
Ir(K) = Z
Gn−r,r
V ol(Kn−r0 )dLn−r[O]
= Z
Gr,n−r
V ol(Kn−r0 )dLr[O],
m(Gn−r,r) = Z
Gn−r,r
dLn−r[O]. For completeness we defineI0(K) =V(K) = volume of K.
The normalization of Ir(K), introduced by Minkowski, is so-called the r-th Minkowski quermassintegrale ofK and is defined by
Wr(K) = (n−r)On−1
nOn−r−1 E(V(Kn−r0 ))
=(n−r)Or−1· · ·O0
nOn−2· · ·On−r−1Ir(K); r= 1,2,· · · , n−1.
For completeness, we put
W0(K) =I0(K) =V(K), Wn(K) =On−1/n.
HereWn−1 is the mean width of the convex bodyK. Those Minkowski quermass- integrales provide powerful tools in theory of convex sets (see [12, 15, 16]).
If Σ is a convex hypersurface bounding a convex bodyKinRn, we have the re- lations between integrals of mean curvatures of Σ(≡∂K) andj-th-order Minkowski quermassintegralsWj ofK (see [12, 15, 16])
Mj(Σ) =nWj+1(K); j= 0,1,· · · , n−1.
Note that Minkowski quermassintegrals Wj are well defined for any convex figure, whereasMj(∂K) makes sense only if∂K is of classC2. We have the following Alexandrov-Fenchelinequalities [16]:
Wi2≥Wi−1Wi+1; W1=A
n; W0=V; Wn =On−1
n ; i= 1,· · ·, n−1,
whereAandV are, respectively, the area and the volume ofK. The equality holds whenK is a standard ball.
3 Geometric Probability and Kinematic Formulas
Consider two compact convex sets A and B in Rn. Let B be fixed and we randomly drop the rigid set A in the space Rn. What is the probability that A
meetsB? In other words, we keepBfixed and letAmoving under the rigid motion g∈Iso(Rn), find the invariant measure of thosegsuch thatB∩gA6=∅. That is, to find the invariant measurem{g∈Iso(Rn)| B∩gA6=∅}. By Hadwiger’s theorem of geometric probability, such an invariant measure equals a linear combination of the n+ 1 intrinsic volumes with coefficients independent of B. We determine these coefficients by taking suitable B’s (for example, letB be an unit ball). This invariant measure is known as a kinematic formula (cf. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]).
LetGbe a Lie group that acts on a smooth manifoldX andH a compact sub- group ofG. Letdgbe an invariant density onG(Haar measure in measure theory).
We consider the invariant measure that comes from the invariant density dg. Let Mp,Nq be two submanifolds of dimensions p, q, respectively, in the homogeneous spaveG/H. We always assume thatM andN are in generic positions. This means that the dimension of the intersectionM∩gN is non-negative for almost allg∈G that is, dim(M ∩gN) =p+q−dim(H/G)≥0. LetI(M∩gN) be an integral in- variant ofM∩gN. One of the basic problems in integral geometry is to find explicit formulas for integral ofI(M ∩gN) overGin terms of known integral invariants of M andN. That is to find numerical constantscnpqi (depends on indices only), such
that Z
G
I(M ∩gN)dg= X
0≤2i≤p+q−n
cnpqiI2i(M)In−2i(N),
whereIk(X) is an invariant ofX. This is called thekinematic formula for I(M ∩gN) For example, let Mp, Nq ⊂ Rn be closed submanifolds with Vol(M) <
∞, Vol(N)< ∞, then Z
{G:M∩gN6=∅}
Vol(M ∩gN)dg=CpqnVol(M)·Vol(N).
(Poincar´e-Blachk´e-Santal´o).
The S.S. Chern’s fundamental formula
LetM, N ⊂Rnbe domains with smooth boundaries∂M, ∂N. LetHj be the j-th mean curvature andχ(M∩gN) be the Euler-Poincar´e characteristics, then we have the following S.S. Chern’s fundamental formula
Z
{G:M∩gN6=∅}
χ(M∩gN)dg=X
i,j
Cijnf(Hj(∂M), Hi(∂N)).
Kinematic formulas are achieved by Shifrin, Federer, Fu, Chen, Howard, Zhang, Zhou and others.
4 Containment Measures and Geometric Inequalities
One geometric question closely related to the isoperimetric inequality ask: given two domainsDk(k=i, j) in the Euclidean spaceRn, when can one domain contain
another? More precisely, we ask if there is an isometry (rigid motion)g ofRn so thatgDj ⊂Di orgDj ⊃Di. We wish to have an answer that depends only on the geometric invariants of domains involved, preferably on the ”surface” areasAlk and the volumesLk (See [7]).
Via method of integral geometry we can obtain the inclusion measure m{g ∈ G : gDj ⊂Di or gDj ⊃ Di} (kinematic formula R
{g∈G:gDj⊂Di or gDj⊃Di}dg).
This measure will lead to many geometric inequalities and their applications. These works are due to G. Zhang and J. Zhou.
Let Dk (k = i, j) be two domains of connected and simply connected and bounded by simple smooth hypersurfaces. LetGbe the group of isometries inRn and dg be the kinematic measure (Harr measure in measure theory) on G. Then we have the following containment measure ([30])
m{g∈G : gDj ⊂Di or gDj ⊃Di}
=m{g∈G : Di∩gDj 6=∅}
−m{g∈G : ∂Di∩g∂Dj6=∅}.
(0.1)
If we can estimate the measurem{g∈G : Di∩gDj6=∅}from below or (and) the measurem{g∈G : ∂Di∩g∂Dj 6=∅}from above in terms of geometric invariants ofDi andDj, then we obtain an inequality of the form
m{g∈G : gDj ⊂Di or gDj ⊃Di}
≥f(A1i,· · ·, Ali;A1j,· · · , Alj),
where each ofAαk (k=i, j;α= 1,· · ·, l) is an integral geometric invariant ofDk. One can then immediately state the following conclusion:
(I). If f(A1i,· · ·, Ali;A1j,· · · , Alj)>0 then there is an isometryg∈Gsuch that eithergDj contains or is contained inDi.
For R2, it is the work of Hadwiger. In the case ofR3 and convex bodies, it is G. Zhang’s work. For more generalRn (n≥3) cases, the work is due to J. Zhou.
(II). If one let Di ≡ Dj ≡ D, then there is no g ∈ G such that gD ⊂ D or gD⊃D. Therefore we have
f(A1(D),· · ·, Al(D))≤0.
This will result in a geometric inequality.
(III). Let Di be, respectively, the in-ball and the out-ball of domainDj (≡D), i.e., the largest ball contained inDand the smallest ball containingD. Then there is nog∈Gsuch that gD⊂Di or gD⊃Di and we have
f(A1(D),· · ·, Al(D), re)≤0;
f(A1(D),· · · , Al(D), ri)≤0.
We would have a Bonnesen-type inequality.
(IV). If we let Di be a ball of radius rbetween the inscribed ball of radius ri
and the circumscribed ball of radiusreofDj (≡D), then we have neithergD⊂Di
norgD⊃Di. Therefore we have an inequality
f(A1(D),· · · , Al(D), r)≤0; ri≤r≤re.
For domains in a plane X²2 of constant curvature² (that is,R2 for²= 0, pro- jective planeP R2for² >0, hyperplane H2 for² <0), these geometric inequalities are investigated in [29] and [30].
Acknowledgement. The first author would like to thank Professor Young Jin Suh for his invitation to theInternational Workshop on Integral Geometry, Submanifolds and Related Topics, which is held in Kyungpook National University during May 25-26, 2007. Most part of this paper comes from the first lecture at Taegu in Korea.
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