Affine function-valued valuations

Jin Li Vienna University of Technology

Convex and Discrete Geometry mathscidoc:1904.40003

International Mathematics Research Notices, 212, 2018
A classification of $\SLn$ contravariant, continuous function-valued valuations on convex bodies is established. Such valuations are natural extensions of $\SLn$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.
Valuation, $\SLn$ contravariant, general projection function, Orlicz projection function, $L_p$ projection function, mixed volume
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  title={Affine function-valued valuations},
  author={Jin Li},
  booktitle={International Mathematics Research Notices},
Jin Li. Affine function-valued valuations. 2018. In International Mathematics Research Notices. pp.212.
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