We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \triangle B\right)$.

**Question:** How can we prove or disprove that, for all $d$-dimensional convex shape $S$, there exists a convex polytope $P$ satisfying simultaneously the following two properties?

Given the constant $\gamma\in (0,1)$ independent of $d$, we have $V_{S,P}\le \gamma V(S)$

The number of its $(d-1)$-dimensional facets $\phi_P$ grows at most polynomially in $d$.

13more comments