Functional Mahler’s conjecture for even s-concave functions (all dimensions)

Establish that for every integer n ≥ 1 and every s > −1/n, for any even s-concave function g: R^n → [0,∞) with 0 < ∫_{R^n} g(x) dx < ∞, the functional volume product ∫_{R^n} g(x) dx · ∫_{R^n} L_s g(y) dy is minimized by the cube indicator function, namely prove ∫_{R^n} g(x) dx · ∫_{R^n} L_s g(y) dy ≥ 4^n / ((1+s)(1+2s)···(1+ns)), with equality for g = 1_{[-1,1]^n}. Here, for s ≠ 0, the s-polar is defined by L_s g(y) = inf_{x: g(x)>0} [(1 − s⟨x,y⟩)_+^{1/s} / g(x)], and for s = 0 by L_0 g(y) = inf_{x: g(x)>0} [e^{−⟨x,y⟩} / g(x)].

Background

Mahler’s classical conjecture for centrally symmetric convex bodies asserts that the volume product |K| * |K^∘| is minimized by the cube. The paper studies a functional analogue where a centrally symmetric convex body is replaced by an even s-concave function g and the polar is replaced by the s-polar L_s g.

The authors define the functional volume product of an even s-concave function g as ∫ g * ∫ L_s g. They propose that this quantity should be minimized by the cube indicator for all s > −1/n, generalizing known cases (e.g., s=0 in dimension 2, and certain unconditional settings).

They prove the conjecture in several low-dimensional or restricted cases but the general form across all dimensions and s > −1/n remains open as stated.