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Minkowski valuations on convex functions.

Andrea Colesanti1, Monika Ludwig2, Fabian Mussnig2

  • 1Dipartimento di Matematica e Informatica "U. Dini", Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Florence, Italy.

Calculus of Variations and Partial Differential Equations
|November 7, 2017
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Summary

This study classifies Minkowski valuations on convex functions and characterizes the projection body operator and LYZ measure. A novel covariant Minkowski valuation is also defined and characterized.

Keywords:
52B45 (26B25, 46B20, 46E35, 52A21, 52A41)

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Area of Science:

  • Convex Geometry
  • Geometric Measure Theory

Background:

  • Minkowski valuations are fundamental tools in convex geometry, extending notions of volume.
  • Understanding operators like the projection body is crucial for analyzing geometric properties of convex bodies.

Purpose of the Study:

  • To classify a specific type of Minkowski valuation (contravariant) on convex functions.
  • To characterize the projection body operator and the associated LYZ measure.
  • To introduce and characterize a new covariant Minkowski valuation.

Main Methods:

  • Utilizing tools from geometric measure theory and the theory of valuations.
  • Developing novel techniques for the analysis of valuations on convex functions.
  • Establishing characterizations through integral geometric formulas.

Main Results:

  • A complete classification of [Formula: see text] contravariant Minkowski valuations on convex functions.
  • A characterization of the projection body operator and the associated LYZ measure.
  • The definition and characterization of a new [Formula: see text] covariant Minkowski valuation.

Conclusions:

  • The established classification provides a deeper understanding of Minkowski valuations.
  • The characterizations offer new insights into the projection body operator and LYZ measure.
  • The newly defined covariant valuation expands the toolkit for geometric analysis.