$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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][1]), the function $f\colon\R\to\R_+$ is log concave. $\quad\Box$

For related results, see e.g. [this paper][2] and references therein. 

  [1]: https://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality#Applications_in_probability_and_statistics
  [2]: https://www-sciencedirect-com.services.lib.mtu.edu/science/article/pii/S0047259X13002534?via%3Dihub