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Positivity of Valuations on Convex Bodies and Invariant Valuations by Linear Actions

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Abstract

In this paper, we endow the space of continuous translation invariant valuations on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with respect to these norms. The novelty of our approach lies in the fact that our proof does not rely on the general theory of wave fronts, but on geometric inequalities deduced from optimal transport methods. We apply this result to prove a variant of Minkowski’s existence theorem, and generalize a theorem of Favre–Wulcan and Lin in complex dynamics over toric varieties by studying the linear actions on the Banach spaces of valuations and by studying their corresponding eigenspaces.

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Notes

  1. For the positivity of forms, we refer the reader to [25, Chapter 3] and [23, Section 1]. In [23, Definition 1.4], “Hermitian positive” is called semipositive.

  2. It was realized in [39] that the same ideas had previously appeared in the classical work of Alexandrov [6].

  3. This can also be obtained by applying Theorem 2.9.

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Acknowledgements

We would like to thank S. Boucksom and C. Favre for their interests and comments regarding this paper. We are particularly grateful to A. Bernig who suggested many improvements of our first result, to T. Wannerer for pointing out a mistake in our previous version. We would also like to thank S. Alesker for answering several questions on his works on the convolution of valuations. The first author would also like to thank L. DeMarco for supporting his stay in Northwestern University to work on this project. We also thank the referee for the careful reading and useful comments.

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  1. Stony Brook University, Stony Brook, USA

    Nguyen-Bac Dang

  2. Tsinghua University, Beijing, China

    Jian Xiao

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  1. Nguyen-Bac Dang
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Correspondence to Nguyen-Bac Dang.

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The first author is supported by the ERC-starting grant project “Nonarcomp” no. 307856, and by the Brazilian project “Ciência sem fronteiras” founded by the CNPq. During the revision of this paper, the second author is supported in part by Tsinghua University Initiative Scientific Research Program (No. 2019Z07L02016) and NSFC (No. 11901336)

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Dang, NB., Xiao, J. Positivity of Valuations on Convex Bodies and Invariant Valuations by Linear Actions. J Geom Anal 31, 10718–10777 (2021). https://doi.org/10.1007/s12220-021-00663-8

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