Entropy-based Bounds on Dimension Reduction in L_1

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 ${\mathrm{\ell }}_1^d$ with distortion $D$ requires dimension $d$ at least $N^{\mathrm{\Omega }(1\mathrm{/}{D}^2)}$, and that for every $\text{eps}>0$ and large enough integer $N$, there exists an $N$-point subset of $L_1$ such that embedding it into ${\mathrm{\ell }}_1^d$ with distortion $1+\text{eps}$ requires dimension $d$ at least $N^{1-O(1\mathrm{/}\log (1\mathrm{/}\text{eps}))}$. 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.