Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is positive almost everywhere and integrable. We know that if $n=1$ and if $f$ is unimodal then the integral $F(x)=\int_{[-\infty,x]} f$ is convex for all $x and concave for all $x>a$, where $a$ is the point of unimodality. Suppose now that one extends the definition of unimodality in the following manner: $\{x\ |\ f(x)\ge a\}$ is a convex set for every $a>0$.
I'm wondering if this result holds as well in the sense that the distribution funciton $F(x_1,\ldots,x_n)=\int_{[-\infty,x_i]} f(x_1,\ldots,x_n)dx_1 \ldots dx_n$ is eventually concave (i.e., in some region $\cap \{x_i\ge t_i\}$.
Intuitively, the restriction on the density function should yield this kind of outcome since as soon as one moves away from the mode, the function starts to decrease. Am I getting the right idea? Can anyone refer me to reference they know of. Thanks in advance.