$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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.

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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.

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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. $\quad\Box$

$\newcommand\u{\mathbf u}\newcommand\v{\mathbf v}\newcommand\x{\mathbf x}\newcommand\R{\mathbb R}\newcommand\ga\gamma$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.