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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The logarithmic Minkowski problem


Authors: Károly J. Böröczky, Erwin Lutwak, Deane Yang and Gaoyong Zhang
Journal: J. Amer. Math. Soc. 26 (2013), 831-852
MSC (2010): Primary 52A40
Published electronically: June 5, 2012
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Abstract: In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite-dimensional Banach space.


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Additional Information

Károly J. Böröczky
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Email: carlos@renyi.hu

Erwin Lutwak
Affiliation: Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, New York 11201
Email: elutwak@poly.edu

Deane Yang
Affiliation: Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, New York 11201
Email: dyang@poly.edu

Gaoyong Zhang
Affiliation: Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, New York 11201
Email: gzhang@poly.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2012-00741-3
PII: S 0894-0347(2012)00741-3
Keywords: Cone-volume measure, Minkowski problem, $L_{p}$-Minkowski problem, log-Minkowski problem
Received by editor(s): September 18, 2011
Received by editor(s) in revised form: February 7, 2012
Published electronically: June 5, 2012
Additional Notes: The research of the first author was supported, in part, by EU FP7 IEF grant GEOSUMSET and OTKA 075016.
The research of the other three authors was supported, in part, by NSF Grant DMS-1007347.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.