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. Then I also have the same question with rotations, i.e., whether $$ h \colon \ [0,2\pi) \to \mathbb{R}_+, \ \alpha \mapsto \mu(\mathbf{u} + \mathbf{R}_\alpha(K-\mathbf{v})) $$ has only one local maximum and one local minimum, where $\mathbf{R}_\alpha$ is a rotation of angle $\alpha$ about some given axis. If these claims are wrong, I am also curious about counterexamples.

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.