### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Journal of Differential Geometry |

Volume | 110 |

Issue number | 1 |

State | Published - Sep 1 2018 |

### Fingerprint

### Keywords

- Aleksandrov problem
- And phrases. Curvature measure
- Integral curvature
- Lp-Aleksandrov problem
- Lp-integral curvature
- Lp-Minkowski problem
- Minkowski problem
- Surface area measure

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

_{P}-Aleksandrov problem for L

_{P}-integral curvature.

*Journal of Differential Geometry*,

*110*(1), 1-29.

**The L _{P}-Aleksandrov problem for L_{P}-integral curvature.** / Huang, Yong; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong.

Research output: Contribution to journal › Article

_{P}-Aleksandrov problem for L

_{P}-integral curvature',

*Journal of Differential Geometry*, vol. 110, no. 1, pp. 1-29.

_{P}-Aleksandrov problem for L

_{P}-integral curvature. Journal of Differential Geometry. 2018 Sep 1;110(1):1-29.

}

TY - JOUR

T1 - The LP-Aleksandrov problem for LP-integral curvature

AU - Huang, Yong

AU - Lutwak, Erwin

AU - Yang, Deane

AU - Zhang, Gaoyong

PY - 2018/9/1

Y1 - 2018/9/1

N2 - It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp 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 Lp-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.

AB - It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp 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 Lp-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.

KW - Aleksandrov problem

KW - And phrases. Curvature measure

KW - Integral curvature

KW - Lp-Aleksandrov problem

KW - Lp-integral curvature

KW - Lp-Minkowski problem

KW - Minkowski problem

KW - Surface area measure

UR - http://www.scopus.com/inward/record.url?scp=85044590321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044590321&partnerID=8YFLogxK

M3 - Article

VL - 110

SP - 1

EP - 29

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -