Optimal constant and extremizers in the TV homogenization inequality

Determine the exact optimal universal constant c* and identify the extremizing families of probability measures that achieve equality in the inequality TV(⊗_{i=1}^n P_i, ⊗_{i=1}^n Q_i) ≥ c* · TV(((1/n)∑_{i=1}^n P_i)^{⊗ n}, ((1/n)∑_{i=1}^n Q_i)^{⊗ n}) for all measurable spaces and all tuples of probability measures (P_1,…,P_n) and (Q_1,…,Q_n).

Background

The paper proves a homogenization inequality for total variation between heterogeneous product measures and their homogenized counterparts, establishing that TV(⊗ P_i, ⊗ Q_i) ≥ c * TV( ( (1/n)∑ P_i )^{⊗ n}, ( (1/n)∑ Q_i )^{⊗ n} ) with an absolute constant c > 0.1489, uniformly over all measurable spaces.

For Bernoulli measures, prior work showed an upper bound c ≤ 8/9 and conjectured that this bound is optimal. The current paper notes that determining the exact optimal constant and the distributions that attain it remains open. This includes identifying whether the conjectured value 8/9 is indeed the optimal universal constant and characterizing extremizing measures.