Suppose I have a simplex $S_n$ with unit side-lengths. Fix a vertex $V$. Let $A_n$ be the convex polytope whose points are contained within the simplex, where the euclidean distance from each point to $V$ is shorter than the distance to any other vertex.
It should be evident that $(n+1)$ copies of $A_n$ can be arranged to form the simplex $S_n$. Therefore, we know that the "volume" of $A_n$ is equivalent to the "volume" of $V_n$ scaled by $\frac{1}{n+1}$. There are still some questions that are bugging me though:
- What is the "surface area" of $A_n$, as a function of $n$ and the "surface area" of $S_n$? I can do calculations for a given $n$, (by using symmetry and adding the "areas" of each individual "face") but I can't seem to find a general formula.
- Does this polytope have a formal name?