Shlomo Reisner

ZONOIDS WITH MINIMAL VOLUME-PRODUCT- A NEW PROOF

A new and simple proof of the following result is given: The product of the volumes of a symmetric zonoid A in RI and of its polar body is minimal if and only if A is the Minkowski sum of n segments.

Shadow Systems and Volumes of Polar Convex Bodies

It is proved that the reciprocal of the volume of the polar bodies, about the Santalo point, of a shadow system of convex bodies K t , is a convex function of t , thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santalo inequality for polytopes in ℝ d that have at most d + 3 vertices.

Local minimality of the volume-product at the simplex

It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of n-dimensional (classes of ) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest, concerning stability of square order of volumes of polars of non-symmetric convex bodies.

Constructing a Polytope to Approximate a Convex Body

We develop an algorithm to construct a convex polytopeP withn vertices, contained in an arbitrary convex bodyK inRd, so that the ratio of the volumes |K/P|/|K| is dominated byc ·. d/n2/(d−1).

Inequalities involving integrals of polar-conjugate concave functions

An inequality of K. Mahler, together with its case of equality, due to M. Meyer, are extended to integrals of powers of polar-conjugate concave functions. An application to estimation of the volume-product of certain convex bodies is given.

The volume of the intersection of a convex body with its translates

It is proved that for a symmetric convex body K in R^n, if for some tau > 0, |K cap (x+tau K)| depends on ||x||_K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies ls are studied.

Hausdorff approximation of convex polygons

We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer (k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min-#, where the goal is to minimize the number of vertices of the output polygon, for a given distance e, and min-e, where the goal is to minimize the error, for a given maximum number of vertices. For min-# problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O(n log n) and O(n) time, for inner and outer approximations, respectively. For min -e problems, the error achieved is within an arbitrary factor α > 1 from the best possible one, and our inner and outer approximation algorithms run in O(f(α, P) ċ n log n) and O (f (α, P) ċ n) time, respectively. Where the factor f (α, P) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P.

Linear time approximation of 3D convex polytopes

We develop algorithms for the approximation of a convex polytope in R3 by polytopes that are either contained in it or containing it, and that have fewer vertices or facets, respectively. The approximating polytopes achieve the best possible general order of precision in the sense of volume-difference. The running time is linear in the number of vertices or facets.

THE CONVEX INTERSECTION BODY OF A CONVEX BODY

Let L be a convex body in n and z an interior point of L . We associate with L and z a new, convex and centrally symmetric, body CI ( L , z ). This generalizes the classical intersection body I ( L , z ) (whose radial function at u ∈ S n −1 is the volume of the hyperplane section of L through z , orthogonal to u ). CI ( L , z ) coincides with I ( L , z ) if and only if L is centrally symmetric about z . We study the properties of CI ( L , z ).

Hausdorff approximation of 3D convex polytopes

Let P be a convex polytope in R, d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous algorithm, where the role of vertices is taken by facets, is presented.