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September 2018 The $L_p$-Aleksandrov problem for $L_p$-integral curvature
Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang
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J. Differential Geom. 110(1): 1-29 (September 2018). DOI: 10.4310/jdg/1536285625


It is shown that within the $L_p$-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural $L_p$ extension, for all real $p$. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the $L_p$-integral curvature of a convex body. This problem is solved for positive $p$ and is answered for negative $p$ provided the given measure is even.

Funding Statement

The first author was supported by the National Science Fund of China for Distinguished Young Scholars (No. 11625103) and the Fundamental Research Funds for the Central Universities of China. The other authors were supported, in part, by USA NSF Grants DMS-1312181 and DMS-1710450.


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Yong Huang. Erwin Lutwak. Deane Yang. Gaoyong Zhang. "The $L_p$-Aleksandrov problem for $L_p$-integral curvature." J. Differential Geom. 110 (1) 1 - 29, September 2018.


Received: 6 November 2015; Published: September 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06933730
MathSciNet: MR3851743
Digital Object Identifier: 10.4310/jdg/1536285625

Primary: 35J20 , 52A38

Keywords: $L_p$-Aleksandrov problem , $L_p$-integral curvature , $L_p$-Minkowski problem , Aleksandrov problem , curvature measure , integral curvature , Minkowski problem , surface area measure

Rights: Copyright © 2018 Lehigh University


Vol.110 • No. 1 • September 2018
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