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.