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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.

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    @LeonidKovalev Sorry - I meant for each $i\in {1,...n}$ $x_i\le t_i$. Do you have any suggestions? If this is not the case, is there some more restricted set, say I restrict x_i < x_j for i< j. Thanks2012-07-24

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This is not true in general. Consider the case where $f$ is the density of a standard 2D Gaussian. Let $\Phi$ be the Gaussian CDF. Then, $F(x, y) = \Phi(x)\Phi(y)$, and some straightforward computations show that at $(0, 0)$ the Hessian of $F$ is indefinite, so it will be neither concave nor convex near there.