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## On the isoperimetric problem in Euclidean space with density Calc (2008)

Venue: | Var. Partial Differential Equations |

Citations: | 40 - 5 self |

### Citations

340 | Lectures on Geometric Measure Theory - Simon - 1983 |

187 |
The Brunn-Minkowski inequality in Gauss space’,
- Borell
- 1975
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Citation Context ...s in manifolds with density has increased. One of the first and most interesting examples, with applications in probability and statistics, is the Gaussian density exp(−π|x| 2 ). About 1975 C. Borell =-=[Bor1]-=- and V. N. Sudakov and B. S. Tirel’son [ST] independently proved that half-spaces minimize perimeter under a volume constraint for this density. In 1982 A. Ehrhard [Eh1] gave a new proof of the isoper... |

88 |
Extremal properties of half-spaces for spherically invariantmeasures. Problems in the theory of probability distributions,
- Sudakov, Tsirel’son
- 1974
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Citation Context ...e of the first and most interesting examples, with applications in probability and statistics, is the Gaussian density exp(−π|x| 2 ). About 1975 C. Borell [Bor1] and V. N. Sudakov and B. S. Tirel’son =-=[ST]-=- independently proved that half-spaces minimize perimeter under a volume constraint for this density. In 1982 A. Ehrhard [Eh1] gave a new proof of the isoperimetric property of half-spaces by adapting... |

69 | Hypersurfaces of constant curvature in space forms.
- Rosenberg
- 1993
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Citation Context ... By (3.1), we get H ′ ψ (0) = nH′ (0) − 〈D X∇ψ, N〉 − 〈∇ψ, D XN〉 = nH ′ (0) − u (∇ 2 ψ)(N, N) + 〈∇ψ, ∇Σu〉 , where in the last equality we have used that DXN = −∇Σu. On the other hand, it is well known =-=[Ro]-=- that (3.5) nH ′ (0) = ∆Σu + |σ| 2 u, where ∆Σ is the Laplacian relative to Σ. Thus, we have obtained H ′ ψ (0) = ∆Σu + |σ| 2 u − u (∇ 2 ψ)(N, N) + 〈∇Σψ, ∇Σu〉 . By substituting this information into (... |

53 | do Carmo, Stability of hypersurfaces with constant mean curvature, - Barbosa, M - 1984 |

50 |
The isoperimetric problem, in “Global theory of minimal surfaces,”
- Ros
- 2005
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Citation Context ...Foundation.2 C. ROSALES, A. CAÑETE, V. BAYLE, AND F. MORGAN Gaussian context. His proof can be simplified by following the generalization of Steiner symmetrization to product measures given by A Ros =-=[R]-=-. More recently, S. Bobkov and C. Houdré [BoH] considered “unimodal densities” with finite total measure in the real line. These authors explicitly computed the isoperimetric profile for such densitie... |

47 |
Some connections between isoperimetric and Sobolev-type inequalities,
- Bobkov, Houdre
- 1997
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Citation Context ...E, AND F. MORGAN Gaussian context. His proof can be simplified by following the generalization of Steiner symmetrization to product measures given by A Ros [R]. More recently, S. Bobkov and C. Houdré =-=[BoH]-=- considered “unimodal densities” with finite total measure in the real line. These authors explicitly computed the isoperimetric profile for such densities and found some of the isoperimetric solution... |

47 | Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115 - Chavel |

44 |
Isoperimetry of waists and concentration of maps, Geom
- Gromov
(Show Context)
Citation Context ...unimodal densities” with finite total measure in the real line. These authors explicitly computed the isoperimetric profile for such densities and found some of the isoperimetric solutions. M. Gromov =-=[Gr]-=- studied manifolds with density as “mm spaces” and mentioned the natural generalization of mean curvature obtained by the first variation of weighted area. V. Bayle [Ba] proved generalizations of the ... |

39 |
Propriétés de concavité du profil isopérimétrique et applications. Thèse de Doctorat, Université Joseph Fourier Grenoble,
- Bayle
- 2004
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Citation Context ...erimetric solutions. M. Gromov [Gr] studied manifolds with density as “mm spaces” and mentioned the natural generalization of mean curvature obtained by the first variation of weighted area. V. Bayle =-=[Ba]-=- proved generalizations of the Lévy-Gromov isoperimetric inequality and other geometric comparisons depending on a lower bound on the generalized Ricci curvature of the manifold. For recent advances o... |

30 |
Extremal properties of half-spaces for log-concave distributions
- Bobkov
- 1996
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Citation Context .... Two relevant examples in probability and statistics where Corollary 4.9 is applied are the standard Gaussian density f(x) = e−πx2 and the logistic density f(x) = e−x (1 + e−x ) −2 . As indicated in =-=[Bo]-=-, for these densities it is also interesting to describe minimizers under a volume constraint of the functionals vol(Ω + [−h, h]) for any h > 0. In [BoH, Remark 13.9] it is pointed out that half-lines... |

21 | Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones,” - Ritore, Rosales - 2004 |

20 |
Symétrisation dans l’espace de
- Ehrhard
- 1983
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Citation Context ...|x| 2 ). About 1975 C. Borell [Bor1] and V. N. Sudakov and B. S. Tirel’son [ST] independently proved that half-spaces minimize perimeter under a volume constraint for this density. In 1982 A. Ehrhard =-=[Eh1]-=- gave a new proof of the isoperimetric property of half-spaces by adapting symmetrization techniques to the Date: February 2, 2008. 2000 Mathematics Subject Classification. 49Q20, 53C17. Key words and... |

19 | Clusters minimizing area plus length of singular curves - Morgan - 1994 |

8 |
Über eine Extremaleigenschaft des Kreises
- Bieberbach
- 1915
(Show Context)
Citation Context ...roduct measures given by A. Ros [R], to construct a counterpart to Steiner symmetrization for the density exp(c|x| 2 ). Then we use this symmetrization in axis directions as employed by L. Bieberbach =-=[Bi]-=- to produce centrally symmetric minimizers with connected boundary. Finally we conclude by Hsiang symmetrization [H] that such a minimizer must be a round ball about the origin. As a corollary of Theo... |

4 |
On the cases of equality in Bobkov's inequality and Gaussian
- Carlen, Kerce
(Show Context)
Citation Context ...f isoperimetric regions is difficult to prove. In the case of the Gaussian density, the complete characterization of equality cases in the isoperimetric inequality is due to E. A. Carlen and C. Kerce =-=[CK]-=-, who proved that any perimeter minimizer for fixed volume is, up to a set of measure zero, a half-space. They obtained this result as consequence of the discussion of equality in a more general funct... |

3 |
extrémaux pour les inégalités de Brunn-Minkowski gaussiennes
- Éléments
- 1986
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Citation Context ...scussion of equality in a more general functional inequality due to S. Bobkov. Previous uniqueness results in the Gaussian setting involving a Brunn-Minkowski type inequality were given by A. Ehrhard =-=[Eh2]-=-. In Theorem 5.2 we also show the uniqueness of round balls centered at the origin as minimizers for the density exp(c|x| 2 ), c > 0. Since round balls appear as the result of finitely many symmetriza... |

1 |
A symmetry theorem on isoperimetric regions
- Hiang
- 1988
(Show Context)
Citation Context ...). Then we use this symmetrization in axis directions as employed by L. Bieberbach [Bi] to produce centrally symmetric minimizers with connected boundary. Finally we conclude by Hsiang symmetrization =-=[H]-=- that such a minimizer must be a round ball about the origin. As a corollary of Theorem 5.2 we deduce an eigenvalues comparison theorem for the density exp(c|x| 2 ), c � 0, generalizing the Faber-Krah... |

1 |
with density
- Manifolds
(Show Context)
Citation Context ...rimetric inequality and other geometric comparisons depending on a lower bound on the generalized Ricci curvature of the manifold. For recent advances on manifolds with density we refer the reader to =-=[M4]-=-, [Ba] and references therein. In this paper we first prove some existence results of isoperimetric regions for densities in Euclidean space with infinite total measure (Theorems 2.2 and 2.6) and reca... |