# Distance function from the origin to the boundary of a convex polytope

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 $(\mathcal S^{n-1},d)$ and define the function $D : \mathcal S^{n-1} \to (0,\infty)$ 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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It will satisfy that for $\alpha=1$. Just consider an arc between $y_1$ and $y_2$ to get to dimension two. There it will be piecewise differentiable with finite left/right derivatives everywhere. –  Lev Borisov Nov 7 '13 at 12:59
The answer to this and other questions can be found in many standard texts on convexity, including Rockafellar, Convex Analysis or Schneider, Convex Bodies: The Brunn-Minkowski theory. –  Deane Yang Nov 7 '13 at 14:43