Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is log concave (density function). Consider now the antiderivative (distribution function) $F(t)=\int_{{ x\le t}}f(x)dx$, which is also log concave. We know that for each component this, say $F_1(\cdot,t_2,\cdots,t_n)$ has regions of concavity and convexity. I was wondering if this result can be extended in the sense that for each point $(t_1,\cdots,t_n)$, either the function $F$ is concave on $\{x \in \mathbb{R}^n|x\ge t\}$ or convex on $\{x \in \mathbb{R}^n|x\le t\}$.
Thanks a bunch in advance for your input.