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## A comparison method for expectations of a class of continuous polytope functionals

• Let $$a_1,\dots,a_n$$ be independent random points in $$\mathbb{R}^d$$ spherically symmetrically but not necessarily identically distributed. Let $$X$$ be the random polytope generated as the convex hull of $$a_1,\dots,a_n$$ and for any $$k$$-dimensional subspace $$L\subseteq \mathbb{R}^d$$ let $$Vol_L(X) :=\lambda_k(L\cap X)$$ be the volume of $$X\cap L$$ with respect to the $$k$$-dimensional Lebesgue measure $$\lambda_k, k=1,\dots,d$$. Furthermore, let $$F^{(i)}$$(t):= $$\bf{Pr}$$ ($$\Vert a_i \|_2\leq t$$), $$t \in \mathbb{R}^+_0$$ , be the radial distribution function of $$a_i$$. We prove that the expectation functional $$\Phi_L$$($$F^{(1)}, F^{(2)},\dots, F^{(n)})$$ := $$E(Vol_L(X)$$) is strictly decreasing in each argument, i.e. if $$F^{(i)}(t) \le G^{(i)}(t)t$$, $$t \in {R}^+_0$$, but $$F^{(i)} \not\equiv G^{(i)}$$, we show $$\Phi$$ $$(\dots, F^{(i)}, \dots$$) > $$\Phi(\dots,G^{(i)},\dots$$). The proof is clone in the more general framework of continuous and $$f$$- additive polytope functionals.