Functional strong slicing conjecture (three variants) for log-concave functions

Determine the sharp upper bounds of the following isotropic constants for integrable log-concave functions f: ℝ^n → ℝ_+: (i) L_f = [max_x f(x) / ∫ f]^{1/n}·(det Cov(f))^{1/(2n)} is maximized by f_0(x) = exp(−∑_{i=1}^n x_i)·1_{[-1,+∞)^n}(x) with value 1; (ii) L̃_f = [ (∫ f)/f(0) ]^{1/n}·(det Cov(f))^{1/(2n)} is maximized by f_1(x) = exp(−∑_{i=1}^n |x_i| ); (iii) L̂_f = e^{−h(f)}·(∫ f)^{1/n}·(det Cov(f))^{1/(2n)} is maximized by f_0(x) with value 1/e.

Background

The authors introduce three natural definitions of an isotropic constant for log-concave functions, differing by the use of max f, f(0), or differential entropy h(f). They conjecture the extremizers and maximal values for each, identifying specific candidate functions f_0 and f_1.

These functional strong slicing conjectures aim to mirror the geometric strong slicing conjectures and, in particular, the entropy-based version (iii) is pivotal for deriving functional Mahler’s conjecture via their main theorem.