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@Sergei Ivanov I believe if $E\subseteq\mathbb{R}^n$ is an ellipsoid centered at the origin then $\mathcal{K}/s^{n+1}$ is constant, where $\mathcal{K}$ is the Gauss curvature of $\partial K$ the boundary of $K$.
@Carl Feynman The support function $h_A:\mathbb{R}^n\to\mathbb{R}$ of a non-empty closed convex set $A$ in $\mathbb{R}^n$ is given by :$ h_A(x)=\sup\{ x\cdot a: a\in A\},$
