Functional Mahler’s conjecture for log-concave functions

Prove that for every integrable log-concave function f: ℝ^n → ℝ_+, the functional volume product P(f) = inf_{z∈ℝ^n} ∫ℝ^n f(x−z) dx · ∫ℝ^n f°(y) dy satisfies P(f) ≥ e^n, with equality for f_0(x) = exp(−∑_{i=1}^n x_i)·1_{[-1,+∞)^n}(x).

Background

The functional version of Mahler’s conjecture parallels the geometric one, replacing convex bodies by integrable log-concave functions and the polar body by the functional polar via the Legendre transform. The Blaschke–Santaló inequality has a functional analogue giving the upper bound, and the conjectured lower bound is en, allegedly attained by a specific extremal log-concave function f_0.

References

The functional form of Mahler's conjecture for log-concave functions postulates that, among log-concave functions P(f) ≥ en, with equality for fo(x) = e-Li=1"(-1,too)n(x). Equivalently, as it happens in the case of convex bodies, it postulates that M(f) ≥ en.

— Entropy, slicing problem and functional Mahler's conjecture (2406.07406 - Fradelizi et al., 11 Jun 2024) in Section 1 (Introduction), after equation (2.1)