On the asymmetry constant of a body with few vertices
Introducing the distance measure \[ d(K,L)=\text{inf}\{\alpha\beta: \alpha>0,\;\beta>0, \frac 1\beta L\subset K\subset \alpha L\} \] for convex bodies \(K,L\subset\mathbb{R}^n\), the authors consider the asymmetry constant \[ \sigma(K) = d(K,C^n) = \text{inf}\{d(K-a,B): a\in\mathbb{R}^n,\;B\subset C^n\} \] of a convex body \(K\subset \mathbb{R}^n\), where \(C^n\) denotes the family of all convex bodies centred at the origin. They are concerned with the problem of how small could be the asymmetry constant of convex \(n\)-dimensional polytopes between (regarding the cardinality of the vertex set) the studied extremal case of a simplex and that of a cross polytope. It is shown that any convex polytope \(P\subset \mathbb{R}^n\) with \(n+k\) vertices, \(1\leq k<n\), is far from any symmetric polytope, and asymptotically sharp estimates for the asymmetry constant of such polytopes are provided.
scientific article; zbMATH DE number 1747819