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I am interested in the following question:

Given a convex body $K$ in $\mathbb{R}^d$ and an $\epsilon>0$ small enough, how many $d$-simplices $\{D_i\}_{i=1}^m\subset K$ do we need at least so that the volume of $K\setminus \cup_{i=1}^m D_i$ is at most $\epsilon\mathrm{vol}(K)$?

Similar question has been considered for approximating polytopes with given number of vertices or faces, but I am not aware of any result of this kind. Any comment shall be greatly appreciated.

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  • $\begingroup$ Don't you need to stipulate that $D_i \subseteq K$? Otherwise one simplex suffices: Take $D_1$ to just completely cover $K$, and then shrink $D_1$ a little. $\endgroup$
    – Joseph O'Rourke
    Nov 10, 2015 at 0:01
  • $\begingroup$ @JosephO'Rourke Yes, of course $D_i$ is contained in $K$. $\endgroup$
    – Roy Han
    Nov 11, 2015 at 1:35

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