Entropy-based bounds on dimension reduction in L1

We show that for every large enough integer N, there exists an N-point subset of L1 such that for every D > 1, embedding it into ℓ1d with distortion D requires dimension d at least $${N^{\Omega (1/{D^2})}}$$, and that for every ɛ > 0 and large enough integer N, there exists an N-point subset of L1 such that embedding it into ℓ1d with distortion 1 + ɛ requires dimension d at least $${N^{\Omega (1/{D^2})}}$$). 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.