2024/12/16 by Matthieu Fradelizi, Fradelizi, Matthieu, Elie Nakhle +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2412.12372
openalex publication_date 2024/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we establish different sharp forms of Mahler's conjecture for s-concave even functions in dimensions n, for n=1 and 2, for s>-1/n, thus generalizing our previous results in \citeFN on log-concave even functions in dimension 2, which corresponds to the case s=0. The functional volume product of an even s-concave function g is ∫ℝng(x)dx∫ℝnLsg(y)dy, where Lsg is the s-polar function associated to g. The analogue of Mahler's conjecture for even s-concave functions postulates that this quantity is minimized for the indicatrix of a cube for any s>-1/n. In dimension n=1, we prove this conjecture for all s∈(-1,0) (the case s≥0 was established by the first author and Mathieu Meyer in \cite[page 17]FM10). In dimension n=2, we only consider the case 1/s∈ℤ: for s>0, we establish Mahler's conjecture for general s-concave even functions; for s<0, the situation is more involved, we only prove a sharp inequality for s-concave functions g such that gs admits an asymptote in every direction. Notice that this set of functions is quite natural to consider, when s<0, since it is the largest subset of s-concave functions stable by s-duality.