Let $K\subseteq \mathbb R^n$, $n\ge 3$ be a convex bounded polytope containing the origin in its interior. Consider the unit sphere with geodesic distance $(\mathcal S^{n1},d)$ and define the function $D : \mathcal S^{n1} \to (0,\infty)$ by $$ D(\gamma) = \sup\{u>0 : u\gamma \in K \}. $$ What can we say about continuity of $D$? In particular, does $D$ satisfy a Hölder condition $D(\gamma_1)  D(\gamma_2) \le C \; d(\gamma_1,\gamma_2)^\alpha$? Can we say something for other convex bodies $K$ ?
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