On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$ -space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.     

Hyperplane sections of convex bodies

We prove sharp inequalities for the volumes of hyperplane sections bisecting a convex body in Rn. This leads to a relative isoperimetric inequality for arbitrary hyperplane sections of a convex body.     

Non-central sections of convex bodies

Let K be a convex body in ? ⁿ . Is K uniquely determined by the areas of its sections? There are classical results that explain what happens in the case of sections passing through the origin. However, much less is known about sections that do not contain the origin. We discuss several problems of this type and establish the corresponding uniqueness results.     

On the maximal volume of convex bodies with few vertices

The problem of determining the largest volume of a (d + 2)-point set in E^d of unit diameter is settled. The extremal polytopes are described completely.     

Gaussian measure of sections of convex bodies

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with the Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.     

Plane sections of centrally symmetric convex bodies

The note contains an example of three plane convex centrally symmetric figuresP ₁,P ₂,P ₃ such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent toP ₁,P ₂,P ₃ respectively.     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

On Dvoretzky’s theorem on almost spherical sections of convex bodies

A complete and rather simple proof of the famous Dvoretzky’s theorem is presented.     

Diameters of sections and coverings of convex bodies

We study the diameters of sections of convex bodies in R^N determined by a random N×n matrix Γ, either as kernels of Γ^* or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R^N has one well bounded k-codimensional section, then for any m>ck random sections of K of codimension m are also well bounded, where c?1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1.     

High dimensional random sections of isotropic convex bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function for random F∈Gn,k and K⊂Rn a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F∈Gn,k with outer volume ratio bounded by

     

Low dimensional sections versus projections of convex bodies

The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝ ⁿ , inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator. Research of this author was partially supported by KBN Grant no. 1 P03A 015 27. This author holds the Canada Research Chair in Geometric Analysis.     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

Modified Busemann-Petty problem on sections of convex bodies

The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ ⁿ with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes of central hyperplane sections so that the answer becomes affirmative in all dimensions. The first-named author was supported in part by the NSF grant DMS-0136022 and by a grant from the University of Missouri Research Board.     

On determinants and the volume of random polytopes in isotropic convex bodies

In this paper we consider random polytopes generated by sampling points in multiple convex bodies. We prove related estimates for random determinants and give applications to several geometric inequalities.     

Generalizations of the Busemann-Petty problem for sections of convex bodies

We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

Ellipsoids of maximal volume in convex bodies

The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterizing ellipsoids of maximal volume contained in convex bodies.