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We prove an exponential deviation inequality for the convex hull of a finitesample of i.i.d. random points with a density supported on an arbitrary convexbody in $\R^d$, $d\geq 2$. When the density is uniform, our result yields rateoptimal upper bounds for all the moments of the missing volume of the convexhull, uniformly over all convex bodies of $\R^d$: We make no restrictions ontheir volume, location in the space or smoothness of their boundary. Afterextending an identity due to Efron, we also prove upper bounds for the momentsof the number of vertices of the random polytope. Surprisingly, these bounds donot depend on the underlying density and we prove that the growth rates that weobtain are tight in a certain sense.

Uniform deviation and moment inequalities for random polytopes with general densities in arbitrary convex bodies