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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: Is a general, explicit characterization of $H(T)$ available?

For instance, the Sudakov minoration yields $$ H(T) \lesssim \bigg(\mathbb{E}_g \sup_{x \in B_2^d} \langle g, Tx\rangle\bigg)^2 \asymp \mathrm{tr}(T^\ast T). $$ (Here $g$ is distributed standard Gaussian in $\mathbf{R}^d$, and $\lesssim$ indicates inequality up to absolute constants and $\asymp$ denotes equality up to constants, independent of everything.) Of course, this cannot be sharp as $T = I_d$ already shows.

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  • $\begingroup$ Do theorems 4 and 5 of Allard and Bolcskei answer your question? $\endgroup$
    – Mark Schultz-Wu
    Commented Nov 20 at 5:37
  • $\begingroup$ Not really, these estimates are also not sharp in general. $\endgroup$
    – Drew Brady
    Commented Nov 20 at 18:45

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