, Volume 13, Issue 1, pp 477-515

Largestj-simplices inn-polytopes

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Abstract

Relative to a given convex bodyC, aj-simplexS inC islargest if it has maximum volume (j-measure) among allj-simplices contained inC, andS isstable (resp.rigid) if vol(S)≥vol(S′) (resp. vol(S)>vol(S′)) for eachj-simplexS′ that is obtained fromS by moving a single vertex ofS to a new position inC. This paper contains a variety of qualitative results that are related to the problems of finding a largest, a stable, or a rigidj-simplex in a givenn-dimensional convex body or convex polytope. In particular, the computational complexity of these problems is studied both for -polytopes (presented as the convex hull of a finite set of points) and for-polytopes (presented as an intersection of finitely many half-spaces).

The research of the first and second authors was supported in part by the Deutsche Forschungs-gemeinschaft and by a joint Max-Planck Research Award. The research of the second author was also supported in part by the Mittag-Leffler Institute and the National Science Foundation.