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Affine Invariant Maps for Log-Concave Functions

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Abstract

Affine invariant points and maps for sets were introduced by Grünbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show that among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santaló point. We also show that the recently introduced floating functions and the John- and Löwner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.

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Acknowledgements

The authors want to thank the referee for the careful reading of the manuscript and the suggestions for improvement.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Ben Li

  2. Mathematisches, Universität Kiel, Kiel, Germany

    Carsten Schütt

  3. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

    Elisabeth M. Werner

  4. Université de Lille 1, UFR de Mathématique, 59655, Villeneuve d’Ascq, France

    Elisabeth M. Werner

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  1. Ben Li
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B. Li: Partially by ERC Grant ERC-770127. E. M. Werner: Partially supported by NSF Grant DMS-1811146 and by a Simons Fellowship.

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Li, B., Schütt, C. & Werner, E.M. Affine Invariant Maps for Log-Concave Functions. J Geom Anal 32, 123 (2022). https://doi.org/10.1007/s12220-022-00878-3

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