A minimax inequality for inscribed cones (Q1260854)

Let \(K\) denote a convex body (non-empty interior) in \(\mathbb{R}^ n\) with Lebesgue measure \(| K |\) and polar \(K^*\). Let \(O\) denote the origin in \(\mathbb{R}^ n\) and set \[ C=\{K \in \mathbb{R}^ n |\;O \in \text{int} K \text{ and } K \text{ is symmetric about } O\}. \] For \(K \in C\) and \(x,y \in \mathbb{R}^ n \backslash \{O\}\), let \(K \langle x,y \rangle\) be the compact cone, inscribed in \(K\), whose base is the intersection of \(K\) with the codimension 1 subspace orthogonal to \(x\), and whose apex is the point of the boundary of \(K\) intersected by the unbounded half line emanating from \(O\) and passing through \(y\). Finally, let \(\omega_ i\) be the \(i\)-dimensional volume of the unit ball in \(\mathbb{R}^ i\). The author establishes the following new minimax inequality: If \(K \in C\) then \[ \min_{x \neq 0} \max_{y \neq 0} | K \langle x,y \rangle | \leq {\omega_ n \omega_{n-1} \over n} | K^*|^{-1}, \] with equality if and only if \(K\) is an ellipsoid.