Comparison Between Plane Symmetric Convex Bodies and Parallelograms.

Random Convex Bodies in a Lattice of Parallelograms

Maximal Parallelograms in Convex Polygons

Given a convex polygon P of n vertices in the plane, we consider the problem of finding the maximum area parallelogram (MAP) inside P. Previously, the best algorithm for this problem runs in time O(n2), and this was achieved by utilizing some nontrivial properties of the MAP. In this paper, we exhibit an algorithm for finding the MAP in time O(n log2 n), greatly improving the previous result. T...

The Cross-section Body, Plane Sections of Convex Bodies and Approximation of Convex Bodies, Ii*

We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, those of sections of convex bodies and of sections of their circumscribed cylinders. For L ⊂ R a convex body, we take n random segments in L and consider their ‘Minkowski average’ D. For fixed n, the pth moments of V (D) (1 ≤ p < ∞) are minimized, for V (L) fixed, by the ell...

Entropy and Asymptotic Geometry of Non–Symmetric Convex Bodies

We extend to the general, not necessary centrally symmetric setting a number of basic results of Local Theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs. Some of these extensions look surprising. The main additional tool is a study of volume behavior around the centroid of the body.

Plane Sections of Convex Bodies of Maximal Volume

Let K = {K0, ...,Kk} be a family of convex bodies in R , 1 ≤ k ≤ n−1. We prove, generalizing results from [9], [10], [13], [14], that there always exists an affine k-dimensional plane Ak ⊆ R , called a common maximal k-transversal of K, such that for each i ∈ {0, ..., k} and each x ∈ R Vk(Ki ∩Ak) ≥ Vk(Ki ∩ (Ak + x)), where Vk is the k-dimensional Lebesgue measure in Ak and Ak + x. Given a famil...