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.



Sliding a convex body over a Gaussian measure
$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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](https://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality#Applications_in_probability_and_statistics)), the function $f\colon\R\to\R\_+$ is log concave. $\quad\Box$


For related results, see e.g. [this paper](https://www.sciencedirect.com/science/article/pii/S0047259X13002534) and references therein.


