### Abstract

For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are "equivalent" in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

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

Pages (from-to) | 1974-1997 |

Number of pages | 24 |

Journal | Advances in Mathematics |

Volume | 231 |

Issue number | 3-4 |

DOIs | |

State | Published - Oct 2012 |

### Fingerprint

### Keywords

- Brunn-Minkowski inequality
- Brunn-Minkowski-Firey inequality
- Minkowski mixed-volume inequality
- Minkowski-Firey L -combinations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*231*(3-4), 1974-1997. https://doi.org/10.1016/j.aim.2012.07.015

**The log-Brunn-Minkowski inequality.** / Böröczky, Károly J.; Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 231, no. 3-4, pp. 1974-1997. https://doi.org/10.1016/j.aim.2012.07.015

}

TY - JOUR

T1 - The log-Brunn-Minkowski inequality

AU - Böröczky, Károly J.

AU - Lutwak, Erwin

AU - Yang, Deane

AU - Zhang, Gaoyong

PY - 2012/10

Y1 - 2012/10

N2 - For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are "equivalent" in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

AB - For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are "equivalent" in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies.

KW - Brunn-Minkowski inequality

KW - Brunn-Minkowski-Firey inequality

KW - Minkowski mixed-volume inequality

KW - Minkowski-Firey L -combinations

UR - http://www.scopus.com/inward/record.url?scp=84864953141&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864953141&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2012.07.015

DO - 10.1016/j.aim.2012.07.015

M3 - Article

VL - 231

SP - 1974

EP - 1997

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 3-4

ER -