A geometric proof of the Blaschke–Lebesgue theorem for the Cheeger constant
[Une preuve géométrique du théorème de Blaschke–Lebesgue pour la constante de Cheeger]
Bogosel, Beniamin1, 2
1 Faculty of Exact Sciences, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Street, Arad (Romania)
2 Centre de Mathématiques Appliquées, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau (France)
Annales de l'Institut Fourier, Online first, 19 p.

The first main result presented in the paper shows that the perimeters of inner parallel sets of planar shapes having a given constant width are minimal for the Reuleaux triangles. This implies that the areas of inner parallel sets and, consequently, the inverse of the Cheeger constant are also minimal for the Reuleaux triangles. Proofs use elementary geometry arguments and are based on direct comparisons between general constant width shapes and the Reuleaux triangle.

Le premier résultat présenté dans cet article montre que le périmètre des ensembles parallèles intérieurs d’une forme de largeur constante est minimal pour le triangle de Reuleaux. Comme conséquence immédiate, le triangle de Reuleaux minimise l’aire d’un ensemble parallèle intérieur et maximise la constante de Cheeger. Les preuves utilisent des arguments de géométrie élémentaire et sont basées sur une comparaison directe entre des formes de largeur constante arbitraires et le triangle de Reuleaux.

Reçu le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3728
Classification : 52A10, 49Q10, 52A38
Keywords: constant width, inner parallel sets, Cheeger constant
Mots-clés : largeur constante, ensemble parallèle intérieur, constante de Cheeger

Bogosel, Beniamin 1, 2

1 Faculty of Exact Sciences, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Street, Arad (Romania)
2 Centre de Mathématiques Appliquées, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau (France) 
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Bogosel, Beniamin. A geometric proof of the Blaschke–Lebesgue theorem for the Cheeger constant. Annales de l'Institut Fourier, Online first, 19 p.

[1] Bezdek, Máté On a generalization of the Blaschke–Lebesgue theorem for disk-polygons, Contrib. Discrete Math., Volume 6 (2011) no. 1, pp. 77-85 | Zbl

[2] Blaschke, Wilhelm Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Math. Ann., Volume 76 (1915) no. 4, pp. 504-513 | DOI | MR | Zbl

[3] Bogosel, Beniamin Numerical Shape Optimization Among Convex Sets, Appl. Math. Optim., Volume 87 (2022) no. 1, 1, 31 pages | DOI | MR | Zbl

[4] Bogosel, Beniamin; Henrot, Antoine; Lucardesi, Ilaria Minimization of the Eigenvalues of the Dirichlet–Laplacian with a Diameter Constraint, SIAM J. Math. Anal., Volume 50 (2018) no. 5, pp. 5337-5361 | DOI | MR | Zbl

[5] Bucur, Dorin; Buttazzo, Giuseppe Variational methods in shape optimization problems, Progress in Nonlinear Differential Equations and their Applications, 65, Birkhäuser, 2005 | DOI | MR | Zbl

[6] Bucur, Dorin; Fragalà, Ilaria; Lamboley, Jimmy Optimal convex shapes for concave functionals, ESAIM, Control Optim. Calc. Var., Volume 18 (2012) no. 3, pp. 693-711 | DOI | Numdam | MR | Zbl

[7] Chakerian, Gulbank D. Sets of constant width, Pac. J. Math., Volume 19 (1966), pp. 13-21 | DOI | MR | Zbl

[8] Cheeger, Jeff A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis. (Papers dedicated to Salomon Bochner, 1969) (Gunning, Robert C., ed.), Princeton University Press (1971), pp. 195-200 | DOI | MR | Zbl

[9] Colesanti, Andrea Brunn–Minkowski inequalities for variational functionals and related problems, Adv. Math., Volume 194 (2005) no. 1, pp. 105-140 | DOI | MR | Zbl

[10] Firey, William J. Isoperimetric ratios of Reuleaux polygons, Pac. J. Math., Volume 10 (1960) no. 3, pp. 823-829 | DOI | MR | Zbl

[11] Gardner, Martin The Unexpected Hanging And Other Mathematical Diversions, Simon & Schuster, Inc., 1969

[12] Ghandehari, Mostafa An optimal control formulation of the Blaschke–Lebesgue theorem, J. Math. Anal. Appl., Volume 200 (1996) no. 2, pp. 322-331 | DOI | Zbl

[13] Harrell, Evans M. II A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal., Volume 12 (2002) no. 1, pp. 81-88 | DOI | MR | Zbl

[14] Henrot, Antoine; Lucardesi, Ilaria A Blaschke–Lebesgue Theorem for the Cheeger constant (2020) | arXiv

[15] Henrot, Antoine; Pierre, Michel Shape variation and optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | DOI | Zbl

[16] Jaglom, Isaak M.; Boltjanskiĭ, Vladimir G. Convex figures, Holt, Rinehart and Winston, 1961 (translated by Paul J. Kelly and Lewis F. Walton) | MR | Zbl

[17] Kawohl, Bernd; Fridman, Vladimir Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant., Commentat. Math. Univ. Carol., Volume 44 (2003) no. 4, pp. 659-667 | MR | Zbl

[18] Kawohl, Bernd; Lachand-Robert, Thomas Characterization of Cheeger sets for convex subsets of the plane, Pac. J. Math., Volume 225 (2006) no. 1, pp. 103-118 | DOI | MR | Zbl

[19] Kupitz, Yaakov S.; Martini, Horst; Perles, Micha A. Ball polytopes and the Vázsonyi problem, Acta Math. Hung., Volume 126 (2010) no. 1-2, pp. 99-163 | DOI | MR | Zbl

[20] Lebesgue, Henri Sur les problèmes des isopérimètres et sur les domaines de largeur constante, Bull. Soc. Math. Fr., Volume 42 (1914), pp. 72-76

[21] Martini, Horst; Montejano, Luis; Oliveros, Déborah Bodies of constant width. An introduction to convex geometry with applications, Birkhäuser, 2019 | DOI | MR | Zbl

[22] Schneider, Rolf Convex bodies: the Brunn–Minkowski theory, Encyclopedia of Mathematics and Its Applications, 151, Cambridge University Press, 2014 | DOI | MR | Zbl

[23] Sokołowski, Jan; Zolesio, Jean-Paul Introduction to shape optimization: shape sensitivity analysis, Springer Series in Computational Mathematics, 16, Springer, 1992 | DOI | MR | Zbl

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