Consider a convex polytope $P$ in $\mathbb{R}^n$ with nonempty interior. Let $r(P)$ denote the inradius of $P$, that is, the radius of the sphere contained in $P$ which touches all facets of $P$, provided such an insphere exists. Let also $\hat{r}(P)$ denote the maximum possible radius of a ball contained in $P$ (note that $\hat{r}(P)$ always exists). The question is, if $r(P)$ exists, is it necessarily true that $r(P)=\hat{r}(P)$?
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Let $X$ be the center of the insphere $S_i$, and $Y$ the center of a maximal sphere $S_m$. Clearly, if $X=Y$ then the spheres' radii are the same.
Suppose that $X\neq Y$. Then there is a nonzero vector $v$ pointing from $X$ to $Y$. If we extend this vector out from $X$ towards infinity, it will cross over at least one bounding hyperplane $B$ of our polytope as it moves from being inside $P$ to being outside of $P$.
But by assumption, the insphere $S_i$ already touches $B$ at some point $b\in B$. So the translation of $S_i$ by $v$ centered at $Y$ (which is contained in $S_m$) passes through the point $b+v$, which is now on the "wrong" side of $B$; this means that the translated sphere exceeds the bounds of our polytope with the given radius, and the maximum radius of a sphere around $Y$ is some value less than $r(P)$. Therefore $r(P)$ must have been the maximum, hence $r(P) = \hat{r}(P)$.
Incidentally, this shows the insphere must be unique if it exists, because we concluded that for any possible insphere center $C$ the inscribed sphere at $C$ has a strictly larger radius than inscribed spheres at any other point $D\neq C$.