The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry

Kai Seng Chou; Xu Jia Wang

doi:10.1016/j.aim.2005.07.004

The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry

Kai Seng Chou^*, Xu Jia Wang

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

295 Citations (Scopus)

Abstract

The L_p-Minkowski problem introduced by Lutwak is solved for p ≥ n + 1 in the smooth category. The relevant Monge-Ampère equation (0.1) is solved for all p > 1. The same equation for p < 1 is also studied and solved for p ∈ (- n - 1, 1). When p = - n - 1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner-type obstruction for this problem is obtained.

Original language	English
Pages (from-to)	33-83
Number of pages	51
Journal	Advances in Mathematics
Volume	205
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2005.07.004
Publication status	Published - 10 Sept 2006

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10.1016/j.aim.2005.07.004

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abstract = "The Lp-Minkowski problem introduced by Lutwak is solved for p ≥ n + 1 in the smooth category. The relevant Monge-Amp{\`e}re equation (0.1) is solved for all p > 1. The same equation for p < 1 is also studied and solved for p ∈ (- n - 1, 1). When p = - n - 1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner-type obstruction for this problem is obtained.",

keywords = "Blaschke-Santalo inequality, Centroaffine Minkowski problem, L-Minkowski problem, Monge-Amp{\`e}re equation",

author = "Chou, {Kai Seng} and Wang, {Xu Jia}",

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AU - Chou, Kai Seng

AU - Wang, Xu Jia

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N2 - The Lp-Minkowski problem introduced by Lutwak is solved for p ≥ n + 1 in the smooth category. The relevant Monge-Ampère equation (0.1) is solved for all p > 1. The same equation for p < 1 is also studied and solved for p ∈ (- n - 1, 1). When p = - n - 1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner-type obstruction for this problem is obtained.

AB - The Lp-Minkowski problem introduced by Lutwak is solved for p ≥ n + 1 in the smooth category. The relevant Monge-Ampère equation (0.1) is solved for all p > 1. The same equation for p < 1 is also studied and solved for p ∈ (- n - 1, 1). When p = - n - 1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner-type obstruction for this problem is obtained.

KW - Blaschke-Santalo inequality

KW - Centroaffine Minkowski problem

KW - L-Minkowski problem

KW - Monge-Ampère equation

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JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -

The L_p-Minkowski problem and the Minkowski problem in centroaffine geometry

Abstract

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