Monotonicity of the classical volume product along heat/Fokker–Planck flow under barycenter conditions

Determine whether, for a nonnegative function f on R^n and its evolution f_t under the heat or Fokker–Planck semi-group, the function t ↦ (∫_{R^n} f_t)(∫_{R^n} (f_t)^{\circ}) is increasing on [0,∞) when either the barycenter of f or the barycenter of its polar function f^{\circ} is at the origin.

Background

The paper proves that a functional L^p volume product M_p(f_t), defined using an infimum over translations of a Laplace-type transform, is monotonically increasing along the heat or Fokker–Planck flow and bounded above by the Gaussian extremal. In the limit p → 0^+, this yields a monotonicity result for the classical functional volume product M(f_t) when the translation is optimized.

However, the authors raise the question of whether one can dispense with the optimizing translation entirely: specifically, they ask whether the direct product (∫ f_t)(∫ (f_t)^{\circ}) increases in time when f or its polar f^{\circ} has barycenter at 0. This would parallel known results for centered functions without requiring re-centering by a Santaló point.