Endre Makai

Maximal sections and centrally symmetric bodies

Let d ≥2 and let K ⊂ℝ d be a convex body containing the origin 0 in its interior. For each direction ω, let the ( d −l)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum when the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on S d−1 , whose null-space needs to be, and will be determined.

Maximal volume enclosed by plates and proof of the chessboard conjecture

and FT, resp., and ~{Vol,_,~~~F;u}~~{Vol,_,F~/F~~v} holds for every UE[W”, then Vol P’IVol P < 2”‘(“-‘). So me related questions are also considered.

INSCRIBING CUBES AND COVERING BY RHOMBIC DODECAHEDRA VIA EQUIVARIANT TOPOLOGY

First, we prove a special case of Knaster’s problem, implying that each symmetric convex body in R 3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S4-equivariant map from SO(3) to S 2 , where S4 acts on SO(3) as the rotation group of the cube and on S 2 as the symmetry group of the regular tetrahedron. We also give some generalizations. Second, we show how the above non-existence theorem yields Makeev’s conjecture in R 3 that each set in R 3 of diameter 1 can be covered by a rhombic dodecahedron, which has distance 1 between its opposite faces. This reveals an unexpected connection between inscribing cubes into symmetric bodies and covering sets by rhombic dodecahedra. Finally, we point out a possible application of our second theorem to the Borsuk problem in R 3 .

ON THE NUMBER OF ANTIPODAL OR STRICTLY ANTIPODAL PAIRS OF POINTS IN FINITE SUBSETS OF R d, II.

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal pairs of points in a subsetX ofRd, which has cardinalityn. The pointsxi, xj∈X are called antipodal if each of them is contained in one of two different parallel supporting hyperplanes of the convex hull ofX. If such hyperplanes contain no further point ofX, thenxi, xj are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for givend andn. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of speciald-simplices, and finally a generalization (concerning strictly antipodal segments) is proved.

Nearly Equal Distances in the Plane

Keywords: distance graph ; subgraph ; points ; Euclidean plane Note: Professor Pachs number: [103]. Also in: Solution to Problem 1 of the 1993 Schweitzer Mathematical Competition, Matematikai Lapok 3 (1993), 31-33 (in Hungarian). Reference DCG-ARTICLE-1993-004doi:10.1017/S0963548300000791 Record created on 2008-11-14, modified on 2017-05-12

Centrally symmetric convex bodies and sections having maximal quermassintegrals

Let

Maximum density space packing with parallel strings of spheres

d \ge 2

On polynomial connections between projections

, and let

On Maximal

K \subset {\Bbb{R}}^d

k

be a convex body containing the origin