Entropy-based Bounds on Dimension Reduction in L₁

We show that for every large enough integer $N$, there exists an $N$-point subset of $L_1$ such that for every $D>1$, embedding it into $\ell_1^d$ with distortion $D$ requires dimension $d$ at least $N^{\Omega(1/D^2)}$, and that for every $\eps>0$ and large enough integer $N$, there exists an $N$-point subset of $L_1$ such that embedding it into $\ell_1^d$ with distortion $1+\eps$ requires dimension $d$ at least $N^{1-O(1/\log(1/\eps))}$. These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman, and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.