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 ℓ ₁ ^d with distortion D requires dimension d at least N^Ω (1/D²), and that for every e{open} > 0 and large enough integer N, there exists an N-point subset of L ₁ such that embedding it into ℓ ₁ ^d with distortion 1 + e{open} requires dimension d at least N^Ω (1/D²)). 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.