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A sharp relative isoperimetric inequality for the square
Comptes Rendus. Mathématique, Volume 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.

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DOI : 10.5802/crmath.243
Haim Brezis 1, 2, 3 ; Alfred Bruckstein 4

1 Dept. of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Depts. of Mathematics and Computer Science,Technion – IIT, Haifa 32000, Israel
3 Laboratoire J.-L. Lions, Sorbonne Universités, UPMC, Université Paris-6, 4 place Jussieu, 75005 Paris, France
4 Dept. of Computer Science, Technion – IIT, Haifa 32000, Israel
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {A sharp relative isoperimetric inequality for the square},
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Haim Brezis; Alfred Bruckstein. A sharp relative isoperimetric inequality for the square. Comptes Rendus. Mathématique, Volume 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 | DOI | MR | Zbl

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[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 | DOI | Zbl

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[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) | MR | Zbl

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