logo CRAS
Comptes Rendus. Mathématique
Equations aux dérivées partielles
A sharp relative isoperimetric inequality for the square
Comptes Rendus. Mathématique, Tome 359 (2021) no. 9, pp. 1191-1199.

We compute the exact value of the least “relative perimeter” of a shape S, with a given area, contained in a unit square; the relative perimeter of S being the length of the boundary of S that does not touch the border of the square.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/crmath.243
@article{CRMATH_2021__359_9_1191_0,
     author = {Haim Brezis and Alfred Bruckstein},
     title = {A sharp relative isoperimetric inequality for the square},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1191--1199},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {9},
     year = {2021},
     doi = {10.5802/crmath.243},
     language = {en},
}
Haim Brezis; Alfred Bruckstein. A sharp relative isoperimetric inequality for the square. Comptes Rendus. Mathématique, Tome 359 (2021) no. 9, pp. 1191-1199. doi : 10.5802/crmath.243. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.243/

[1] Yaniv Altshuler; Alfred M. Bruckstein On Short Cuts or Fencing in Rectangular Strips (2010) (https://arxiv.org/abs/1011.5920)

[2] Luigi Ambrosio; Jean Bourgain; Haim Brezis; Alessio Figalli BMO-Type Norms Related to the Perimeter of Sets, Commun. Pure Appl. Math., Volume 69 (2016) no. 6, pp. 1062-1086 | Zbl 1352.46026

[3] Dominique Bakry; Michel Ledoux Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math., Volume 123 (1996) no. 2, pp. 259-281 | Zbl 0855.58011

[4] Franck Barthe; Bernard Maurey Some remarks in isoperimetry of Gaussian type, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 36 (2000) no. 4, pp. 419-434 | Zbl 0964.60018

[5] Sergey G. Bobkov A functional form of the isoperimetric inequality for the Gaussian measure, J. Funct. Anal., Volume 135 (1996) no. 1, pp. 39-49 | Zbl 0838.60013

[6] Sergey G. Bobkov An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab., Volume 25 (1997) no. 1, pp. 206-214 | Zbl 0883.60031

[7] Hugo Hadwiger Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math., Volume 76 (1972), pp. 410-418 | Zbl 0248.52012

[8] Hugh Howards; Michael Hutchings; Frank Morgan The Isoperimetric Problem on Surfaces, Amer. Math. Monthly, Volume 106 (1999) no. 5, pp. 430-439 | Zbl 1003.52011

[9] Wladyslaw Kulpa The Poincaré–Miranda theorem, Amer. Math. Monthly, Volume 104 (1997) no. 6, pp. 545-550 | Zbl 0891.47040

[10] Jean Mawhin Simple proofs of the Hadamard and Poincaré–Miranda theorems using the Brouwer fixed point theorem, Amer. Math. Monthly, Volume 126 (2019) no. 3, pp. 260-263 | Zbl 1415.54025

[11] Henri Poincaré Sur certaines solutions particulières du problème des trois corps, Bull. Astronomique, Volume 1 (1884), pp. 65-74

[12] Antonio Ros The isoperimetric problem, Global theory of minimal surfaces (David Hoffman, ed.) (Clay Mathematics Proceedings), Volume 2, American Mathematical Society, 2005, pp. 175-209 (Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25–July 27, 2001) | Zbl 1125.49034

Cité par document(s). Sources :