Extremal Spheres and Semi-Infinite Duality Theory

Abstract

Let $\mathbb{K}$ be the class of all nonempty compact convex sets of the n-dimensional Euclidean vector space ℝⁿ. For K ∈ $\mathbb{K}$ we denote the inradius ϱ and the circumradius R by

$$\varphi = \varphi (K): = \sup \{ r/B(x,r) \subseteq K\} $$

((1))

,

$$R = R(K): = \inf \{ r/B(x,r) \supseteq K\} $$

((2))

,where $B(x,r): = \{ t \in {\mathbb{R}^n}/\left\| {t - x} \right\| \leqslant r\}$.