Functional slicing conjecture (weak form) for log-concave functions

Determine whether there exists a universal constant C, independent of n and f, such that L̂_f = e^{−h(f)}·(∫ f)^{1/n}·(det Cov(f))^{1/(2n)} ≤ C for every integrable log-concave function f: ℝ^n → ℝ_+.

Background

Analogous to the geometric slicing conjecture, the functional slicing conjecture asks for a dimension-free bound on a suitably defined isotropic constant for log-concave functions. The authors emphasize that among the candidate definitions, the entropy-based isotropic constant L̂_f is the one that aligns with their main theorem linking strong slicing to Mahler’s conjecture in the functional setting.

References

The usual slicing conjecture for log-concave function asks if their isotropic constants are upper bounded by an absolute universal constant, while its stronger form postulates that, in any fixed dimension n, for any log-concave function f on R n , one has L f ≤ L f0 = 1/e.

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