Lower bound for the functional volume product among s-concave functions with g^s in F (s<0)

Determine, for every integer n ≥ 1 and s ∈ (−1/n, 0), whether for any even function g: R^n → [0,∞) with g^s ∈ F the lower bound ∫_{R^n} g(x) dx · ∫_{R^n} L_s g(y) dy ≥ (1/(1+ns)) · 4^n / ((1+s)(1+2s)···(1+ns)) holds, with equality for g(x) = (max(1, ||x||_∞))^{1/s}. Here F denotes the class of convex lower semi-continuous functions f: R^n → (0,∞) such that for any x ≠ 0, lim_{t→∞} f(tx) = ∞ and t ↦ f(tx)/t is non-increasing on (0,∞), and L_s g 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, the s-dual of an integrable s-concave function has additional structure: (L_s g)^s admits an asymptote in every direction, motivating the introduction of the class F of convex lower semi-continuous functions with specific growth and monotonicity along rays.

The conjecture proposes the sharp lower bound for the functional volume product ∫ g * ∫ L_s g within the subclass of even s-concave functions satisfying g^s ∈ F, identifying a specific extremizer g(x) = (max(1, ||x||_∞))^{1/s}.

The authors verify this bound in certain cases (e.g., unconditional functions, dimension 2 with −1/s ∈ N), but the general case across all dimensions remains open.