Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

On a Problem in Combinations

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

On general construction for extended visual cryptography schemes

Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.

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Extremal problems for convex polygons

Visual cryptography schemes have been introduced in 1994 by Naor and Shamir. Their idea was to encode a secret image into n shadow images and to give exactly one such shadow image to each member of a group P of n persons. Whereas most work in recent years has been done concerning the problem of qualified and forbidden subsets of P or the question of contrast optimizing, in this paper we study extended visual cryptography schemes, i.e., shared secret systems where any subset of P shares its own secret.

Extended visual cryptography schemes

The largest small n -dimensional polytope with n + 3 vertices

A correction to The largest small n -dimensional polytope with n + 3 vertices

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

A geometric version of dyson's lemma for factors of arbitrary dimension

In this paper, we present a generalization of a result due to Hollmann, Korner, and Litsyn [9]. They prove that each partition of the n-dimensional binary Hamming space into spheres consists of either one or two or at least n + 2 spheres. We prove a q-ary version of that gap theorem and consider the problem of the next gaps. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 183–201, 2006

Markus Wessler

Papers

Extended visual cryptography schemes

The largest small n -dimensional polytope with n + 3 vertices

A correction to The largest small n -dimensional polytope with n + 3 vertices

A geometric version of dyson's lemma for factors of arbitrary dimension

On partitions of the q-ary Hamming space into few spheres