# Log-concavity of areas of level sets

Suppose $$f: \mathbb{R}^d \to \mathbb{R}$$ is a smooth convex function.

Consider the level sets of the function, namely $$M_s = \{x: f(x) = s\}$$. Is it true/known that the surface areas of $$M_s$$ are log-concave as a function of $$s$$?

(This feels awfully like a Brunn-Minkowski style inequality, but I'm unsure if it follows from known results. If so, references are highly appreciated!)

• It is true for smooth convex radial functions. Commented Oct 30, 2018 at 1:48
• Indeed, that's where my intuition comes from that this may be true. Commented Oct 30, 2018 at 8:08

Yes, this is true, and you are right, this follows from a generalization of the Brunn-Minkowski inequality.

Let $$K_s = \{x \mid f(x) \le s\}$$, so that $$M_s = \partial K_s$$. We have $$K_s \supseteq (1-s)K_0 + sK_1$$, thus the surface area of the former is $$\ge$$ the surface area of the latter.

The surface area of a convex body can be written as a mixed volume: $$\mathrm{vol}_{n-1}(\partial K) = n \mathrm{vol}(\underbrace{K, \ldots, K}_{n-1}, B).$$ A general version of the Brunn-Minkowski inequality says that the function $$\mathrm{vol}(\underbrace{(1-t)K_0 + tK_1, \ldots, (1-t)K_0 + tK_1}_{n-i}, L_1, \ldots, L_i)^{\frac1{n-i}}$$ is concave for any convex bodies $$L_1, \ldots, L_i$$.

It follows that the $$(n-1)$$-st root of the surface area of $$K_t$$ is concave. Concavity of $$f^{\frac{1}{n-1}}$$ implies concavity of $$\log f$$, and we are done.

References:

Gardner, R. J., The Brunn-Minkowski inequality, Bull. Am. Math. Soc., New Ser. 39, No. 3, 355-405 (2002). ZBL1019.26008, Section 17.

Burago, Yu. D.; Zalgaller, V. A., Geometric inequalities. Transl. from the Russian by A. B. Sossinsky, Grundlehren der Mathematischen Wissenschaften, 285. Berlin etc.: Springer-Verlag. XIV, 331 p.; DM 184,- (1988). ZBL0633.53002, p. 146.

Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001, Theorem 6.4.3.