John's Position is not good for approximation

2017/03/07 by Huang, Han
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1703.02173

Abstract

Recall that a convex body K is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in K. Approximating convex body in John's position by polytopes we obtain the following results. 1. Let n>R_n≥ 1 be a sequence such that lim_{n→ ∞} (R_n)/(n)=0. For a sufficiently large n, we can construct a convex body K⊂ ℝⁿ in John's position such that there is no P, polytope with a polynomial number of facets in n such that K⊂ P⊂ R_nK; 2. For a sufficiently large n, there is a convex body K⊂ ℝⁿ in John's position such that there is no P, polytope that has less than exp(cn) facets satisfies K⊂ P ⊂ √(n)K.

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