Question

Depth is defined as the distance to the boundary of a set, i.e., $\operatorname{depth}(x, C) = \operatorname{dist}(x, \mathbb{R}^n \backslash C)$. Let $C$ be a convex set that contains the origin. I believe that $$\operatorname{depth}(0, C) = \min_{\|u\| = 1} \ \max_{v \in C} \ u \cdot v$$ I also believe the following lemma is true: given $u^*$ that optimizes the above expression, $\max u^* \cdot v$ is achieved at $v = \lambda u^*$ for some scalar $\lambda$ (though not necessarily uniquely).

I don't know how to prove these things or where to look in the literature for these sorts of results.

Answer 1

My office is quieter this morning, so let's try again. I'll assume that $C$ is compact with $0 \in \operatorname{int} C$. Let $B_r$ denote the ball of radius $r$ centered at $0$. Then $\operatorname{depth} (0,C) = \max \{ r > 0 \mid B_r \subseteq C \}$.

The function $h_C(u) = \max_{v\in C} (u \cdot v)$ is called the support function of $C$. Its crucial property here is that given two compact convex sets $C$ and $K$, $h_C \le h_K$ if and only if $C \subseteq K$.

On the one hand, if $B_r \subseteq C$ then for every unit vector $u$, $h_C(u) \ge h_{B_r}(u) = r$, so $$\operatorname{depth} (0,C) \le \min_{\| u \| = 1} h_C(u).$$ On the other hand, if $h_C(u) \ge r$ for every unit vector $u$, then $h_C \ge h_{B_r}$ and so $B_r \subseteq h_C$, and thus $$\operatorname{depth} (0,C) \ge \min_{\| u \| = 1} h_C(u).$$

As for the second question, if $r = \operatorname{depth}(0,C)$ and $u^\ast$ achieves the minimum, then $r u^\ast \in \partial B_r \cap \partial C$. Since $u^\ast \cdot (r u^\ast) = r$, the maximum is achieved for $v = r u^\ast \in C$.