All Questions

Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map. Consider the function $$ H(T) := \log M(TB_2^d, B_2^d), $$ which is the packing entropy for $TB_2^d$ by $B_2^d$....

Question: What is the smallest number of nonoverlapping unit cubes that can hide a unit cube C - in the sense that every ray emanating from the boundary of C meets the interior or the boundary of one ...

Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...

Take a standard simplex or cube in $\mathbb R^n$. Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities? If not what ...

Is there a bounded convex polytope $\mathcal P\subseteq\mathbb R^n$ with $m$ vertices, whose vertex vectors span $\mathbb R^n$ (so $m$ is $\Omega(n)$) and just $O(poly(\log n))$ half-plane ...

Simply put, my question is this: what is the smallest dimension, if any, where we can know for sure that a convex body exists whose translative packing constant is strictly larger than its lattice ...