Finding Largest Simplices (Preliminary Report)

Abstract

Many of the basic problems of computational geometry are well understood only in low-dimensional cases. That is especially true of most geometric optimization problems except those that can be directly transformed into problems of linear or concave maximization. In the case of unrestricted dimension, nonconcave geometric maximization problems provide a fertile testing ground for methods of global optimization, for such problems typically present many local maxima that are not global maxima, and even the local maxima may be hard to find and recognize. That is often true even of problems involving the simplest sorts of convex polytopes (parallelotopes, cubes, simplices, etc.). As an indication of the current state of knowledge (or ignorance!) concerning high-dimensional geometric optimization problems, we here survey the fragmentary situation with respect to the following two problems concerning largest simplices (where k and d are integers with 1 ≤ k ≤ d).