# zbMATH — the first resource for mathematics

Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. (Russian) Zbl 0648.52003
The following extremal problem is investigated. Let $$\rho$$ be a measure over $${\mathbb{R}}^ n:$$ find, among all (centered at the origin) oblic parallelepipeds with prescriped heights the one with a minimal $$\rho$$- volume. It is the main aim of the paper to show that, for a large class of spherically symmetric measure, a solution of this problem is to be obtained with orthogonal parallelepipeds. Applications are given to the evaluation of some characteristic invariants of convex polyhedra.
Reviewer: M.Turinici

##### MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 28A75 Length, area, volume, other geometric measure theory 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52Bxx Polytopes and polyhedra
Full Text: