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Register a new userOn extendability by continuity of valuations on convex polytopes
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Authors
Semyon Alesker
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There is a well known construction of weakly continuous valuations on convex compact polytopes in R^n. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of R^n. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R^3 more explicit results are obtained.
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The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the authors irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.
The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.
Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbatys theorem to obtain a few apparently new inequalities for mixed volumes of convex bodies.
Let $K$ be a convex body in $mathbb{R}^n$ and $f : partial K rightarrow mathbb{R}_+$ a continuous, strictly positive function with $intlimits_{partial K} f(x) d mu_{partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
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