Comparison inequality between a function and its double s-polar (s<0)

Establish that for every integer n ≥ 1 and s ∈ (−1/n, 0), for any even s-concave function g: R^n → [0,∞) with 0 < ∫_{R^n} g(x) dx < ∞, the integral of g dominates that of its double s-polar up to the factor (1+ns), namely prove ∫_{R^n} g(x) dx ≥ (1+ns) · ∫_{R^n} L_s L_s g(x) dx, with equality for g = 1_{[-1,1]^n}. Here L_s is the s-polar defined by L_s g(y) = inf_{x: g(x)>0} [(1 − s⟨x,y⟩)_+^{1/s} / g(x)].

Background

For s<0, while one always has the pointwise bound g(x) ≤ L_s L_s g(x), the authors conjecture an opposite integral inequality (up to a known constant) for even s-concave functions. This forms a key component in deducing the full functional Mahler conjecture from the specialized lower bound in the class F.

They verify the conjecture in dimension n=1, and show how, together with the conjectured bound in the F-class, it would imply the full functional Mahler conjecture for s<0.