A norm - inequality related to affine regular hexagons

Let $(E, lVert . rVert)$ be a two-dimensional real normed space with unit sphere $S = {x in E, lVert x rVert = 1}$. The main result of this paper is the following: Consider an affine regular hexagon with vertex set $H = {pm v_1, pm v_2, pm v_3} subseteq S$ inscribed to $S$. Then we have $$min_i max_{x in S}{lVert x - v_i rVert + lVert x + v_i rVert} leq 3.$$ From this result we obtain $$min_{y in S} max_{x in S}{lVert x - y rVert + lVert x + y rVert} leq 3,$$ and equality if and only if $S$ is a parallelogram or an affine regular hexagon.

On the gap of finite metric spaces of p-negative type

Let (X,d) be a metric space of p-negative type. Recently I. Doust and A. Weston introduced a quantification of the p-negative type property, the so called gap {Gamma} of X. This talk introduces some formulas for the gap {Gamma} of a finite metric space of strict p-negative type and applies them to evaluate {Gamma} for some concrete finite metric spaces.