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!)



Log-concavity of areas of level sets
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**](http://dx.doi.org/10.1090/S0273-0979-02-00941-2), Bull. Am. Math. Soc., New Ser. 39, No. 3, 355-405 (2002). [ZBL1019.26008](https://zbmath.org/?q=an:1019.26008), 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](https://zbmath.org/?q=an:0633.53002), 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](https://zbmath.org/?q=an:0798.52001), Theorem 6.4.3.


