Functional strong slicing conjecture (even functions)

For integrable log-concave even functions f: ℝ^n → ℝ_+, establish the sharp upper bounds: (i) L_f = [max_x f(x) / ∫ f]^{1/n}·(det Cov(f))^{1/(2n)} is maximized by f_1(x) = exp(−∑_{i=1}^n |x_i| ); (ii) L̂_f = e^{−h(f)}·(∫ f)^{1/n}·(det Cov(f))^{1/(2n)} is maximized by the indicator of the hypercube, f(x) = 1_{[-1,1]^n}(x).

Background

Restricting to even log-concave functions simplifies certain relationships (e.g., max f = f(0)), and the authors formulate symmetric analogues of the functional strong slicing conjecture. These conjectures align with known sharp results in dimension one and propose specific extremizers in higher dimensions.

This even-function setting is the functional counterpart to the symmetric strong slicing conjecture for convex bodies.