On intersections of convex bodies

Let $K_0,K_1,\dots,K_m$ be nonempty convex bodies in $\mathbb R^n$. Let $r_1,\dots,r_m$ be vectors in $\mathbb R^n$, $\rho=(r_1,\dots,r_m)\in\mathbb R^{nm}$. Then the set $D=\{\rho\mid\Phi(\rho)K_0\cap\bigcap^m_{i=1}(K_i+r_i)\ne\varnothing\}$ is convex in $\mathbb R^{nm}$, and the family of sets $\{\Phi(\rho)\mid\rho\in D\}$ is concave. Let $k=\max\limits_\rho\dim\Phi(\rho)\ge1$. Then for the volume $\operatorname{Vol}_{k}\Phi(\rho)=W_0(\Phi(\rho))$ and for all mean cross-sectional measures $W_\nu(\Phi(\rho))$, $\nu=0,1,\dots,k-1$, the function $\sqrt[k-\nu]{W_\nu(\Phi(\rho))}$ is concave on the set $D$.