Sliding a convex body over a Gaussian measure

Question

Consider an $n$-dimensional convex set $K \subset \mathbb{R}^n$ and let $\mu$ denote the Gaussian measure with density $$ \gamma(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}} e^{-\lVert \mathbf{x} \rVert^2/2}. $$ I am trying to figure out if, as I slide the convex body $K$ along a straight line, its Gaussian measure, viewed as a function of the line parameter, is a function that has a unique local and global maximum. In essence, I want to know if $$ g \colon \ \mathbb{R} \to \mathbb{R}_+, \ t \mapsto \mu(\mathbf{u} + t\mathbf{v} + K) $$ is log-concave or quasi-concave.

Answer 1

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$We have $$\mu(\u+t\v+K)=f(t):=\int_{\R^n}d\x\,F(\x,t),$$ where $$F(\x,t):=\ga(\x)\,1(\x\in\u+t\v+K).$$ The function $F\colon\R^n\times\R\to\R_+$ is log concave, as the product of two log-concave functions. So, by the Prékopa–Leindler inequality (cf. e.g. this), the function $f\colon\R\to\R_+$ is log concave. $\quad\Box$

For related results, see e.g. this paper and references therein.