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**Setup:**


Let $C\_n$ be a closed $n$-simplex in $\mathbb{R}^n$ and let $r \in (0,R)$ where $R$ is the distance any one of the vertices $\{v\_1,\cdots , v\_{n+1}\}$ of $C\_n$ to the centroid $\frac{v\_1+ \cdots v\_{n+1}}{n+1}\in C\_n$. 




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**Question:**


Is there a way or removing a connected open set $A$ from the interior of $C\_n$ such that for every $c \in \partial C\_n$
$$
r= d(\partial A, c) \left(\triangleq \inf\_{a \in \partial A}d(a,c)\right)?
$$



Isometry between punctured sphere and punctured triangle?
There is no such $A$. Let $c$ be a point from the boundary of $C\_n$ that belongs to an intersection of at least two distinct facets $F\_1, F\_2$ (so $c$ is in a face of dimension at most $n-2$). In the plane, $c$ is a vertex of the triangle $C\_2$.


Since the simplex is regular, the angle between any two of its facets is sharp ( smaller than $\pi/2$). In particular, if $x$ is a point in $C\_n$ at distance $r+\varepsilon$ from $c$, then both $d(F\_1,x)$ and $d(F\_2,x)$ are strictly smaller than $(r+\varepsilon)/\sqrt{2}$.


