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arXiv · 2005.00486

The support of dually epi-translation invariant valuations on convex functions

Jonas Knoerr
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Abstract

We study dually epi-translation invariant valuations on cones of convex functions containing the space of finite-valued convex functions. The existence of a homogeneous decomposition is used to associate a distribution to every valuation of this type similar to the Goodey-Weil embedding for translation invariant valuations on convex bodies. The relation between the valuation and its associated distribution is used to establish a notion of support for valuations. As an application, we show that there are no $\mathrm{SL}(n)$ or translation invariant valuations except constant valuations in this class and we discuss which valuations on finite-valued convex functions can be extended to larger cones. In addition, we examine some topological properties of spaces of valuations with compact support.

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Jonas Knoerr. 2020-04-30. The support of dually epi-translation invariant valuations on convex functions. https://doi.org/10.1016/j.jfa.2021.109059

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