In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 ≤ α ≤ β, called (α, β)-convex bodies. We show that among planar (α, β)-convex bodies of fixed perimeter, the extremal shape is a domain whose boundary is composed by two arcs of circles of radius α joined by two arcs of circles of radius β.
@article{COCV_2022__28_1_A62_0,
author = {Croce, Gisella and Fattah, Zakaria and Pisante, Giovanni},
title = {A reverse isoperimetric inequality for planar $( \alpha , \beta )$-convex bodies},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022056},
mrnumber = {4487866},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022056/}
}
TY - JOUR AU - Croce, Gisella AU - Fattah, Zakaria AU - Pisante, Giovanni TI - A reverse isoperimetric inequality for planar $( \alpha , \beta )$-convex bodies JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022056/ DO - 10.1051/cocv/2022056 LA - en ID - COCV_2022__28_1_A62_0 ER -
%0 Journal Article %A Croce, Gisella %A Fattah, Zakaria %A Pisante, Giovanni %T A reverse isoperimetric inequality for planar $( \alpha , \beta )$-convex bodies %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022056/ %R 10.1051/cocv/2022056 %G en %F COCV_2022__28_1_A62_0
Croce, Gisella; Fattah, Zakaria; Pisante, Giovanni. A reverse isoperimetric inequality for planar $( \alpha , \beta )$-convex bodies. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 62. doi: 10.1051/cocv/2022056
[1] and , Control theory from the geometric viewpoint, volume 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004). Control Theory and Optimization, II. | MR
[2] , and , Optimal control. Contemporary Soviet Mathematics. Consultants Bureau, New York (1987). Translated from the Russian by . | MR | Zbl
[3] , Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. 44 (1991) 351-359. | MR | Zbl | DOI
[4] , Optimisation de forme dans la classe des corps de largeur constante et des rotors. Theses, Université Pierre et Marie Curie - Paris VI (2007).
[5] , Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory. SIAM J. Control Optim. 47 (2009) 3007-3036. | MR | Zbl | DOI
[6] and , Objets convexes de largeur constante (en 2D) ou d’épaisseur constante (en 3D): du neuf avec du vieux. Ann. Sci. Math. Québec 36 (2013) 17-42. | Zbl | MR
[7] , A reverse isoperimetric inequality and its application to the gradient flow of the Helfrich functional (2020). [math.AP] | arXiv | Zbl
[8] and , Isoperimetric inequality for curves with curvature bounded below. Math. Notes 95 (2014) 590-598. | MR | Zbl | DOI
[9] , and , A sausage body is a unique solution for a reverse isoperimetric problem. Adv. Math. 353 (2019) 431-445. | MR | Zbl | DOI
[10] , On reduced convex bodies. Israel J. Math. 56 (1986) 247-256. | MR | Zbl | DOI
[11] , On a solution of the reverse Dido problem in a class of convex surfaces of revolution. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 4 (2016) 7-12. | MR | Zbl
[12] , and , A quantitative version of the isoperimetric inequality: the anisotropic case. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4 (2005) 619-651. | MR | Zbl | Numdam
[13] and , Blaschke-Santalé diagram for volume, perimeter, and first Dirichlet eigenvalue. SIAM J. Math. Anal. 53 (2021) 1670-1710. | MR | Zbl | DOI
[14] , Reverse isoperimetric inequalities in R3. ProQuest LLC, Ann Arbor, MI (2012). Ph.D. thesis, The Ohio State University. | MR
[15] and , editors, Handbook of convex geometry. Vol. A, B. North-Holland Publishing Co., Amsterdam (1993). | Zbl
[16] , Notions of convexity, Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA (2007). Reprint of the 1994 edition. | MR | Zbl
[17] , Convex bodies of constant width and constant brightness. Adv. Math. 204 (2006) 241-261. | MR | Zbl | DOI
[18] and , A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature. Rocky Mountain J. Math. 25 (1995) 635-684. | MR | Zbl | DOI
[19] , Generalized curvatures, volume 2 of Geometry and Computing. Springer-Verlag, Berlin (2008). | MR | Zbl
[20] , A reverse quantitative isoperimetric type inequality for the Dirichlet Laplacian (2021). [math.AP] | arXiv | MR | Zbl
[21] , Integral geometry and geometric probability. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition (2004). With a foreword by Mark Kac. | MR | Zbl
[22] , Convex bodies: the Brunn-Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, expanded edition (2014). | MR
[23] , Singular trajectories and subanalyticity in optimal control and Hamilton-Jacobi theory. Rend. Semin. Mat. Univ. Politec. Torino 64 (2006) 97-109. | MR | Zbl
[24] , Convex sets. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Toronto-London (1964). | MR | Zbl
[25] , Inequalities for curvature integrals in Euclidean plane. J. Inequalit. Appl. 2019 (2019) 1-12. | MR | Zbl
Cité par Sources :
