The missing (A,D,r) diagram
Annales de l'Institut Fourier, Online first, 52 p.

In this paper we are interested in “optimal” universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term “optimal” is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke–Santaló diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A,D,r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears to be quite elementary, it had not been answered until now. This is probably related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.

Dans cet article, nous nous intéressons à des inégalités géométriques universelles “optimales” mettant en jeu l’aire, le diamètre et l’inradius des corps convexes. Le terme“optimal” doit être compris dans le sens suivant  : nous étudions la question de la minimisation/maximisation de la mesure de Lebesgue d’un corps convexe parmi tous les ensembles convexes de diamètre et de rayon intérieur donnés. Le problème de minimisation dans le cas bidimensionnel a été résolu dans un article précédent, par M. Hernandez–Cifre et G. Salinas. Dans cet article, nous fournissons une généralisation au cas n-dimensionnel basée sur une approche différente, ainsi que la résolution complète du problème de maximisation dans le cas bidimensionnel. Cela nous permet de déterminer complètement le diagramme de Blaschke–Santaló bidimensionnel pour les corps convexes planaires par rapport aux trois grandeurs aire, diamètre et inradius dans les espaces euclidiens, noté (A,D,r). Un tel diagramme est utilisé pour déterminer la gamme des valeurs possibles de l’aire des ensembles convexes en fonction de leur diamètre et de leur inradius. Bien que cette question de géométrie convexe semble assez élémentaire, elle n’avait pas trouvé de réponse jusqu’à présent. Ceci est probablement lié au fait que la description du diagramme utilise des ensembles convexes particuliers inattendus, tels qu’une sorte de nonagone lissé inscrit dans un triangle équilatéral.

Online First:
DOI: 10.5802/aif.3484
Classification: 49Q10,  52A40,  28A75,  49K15
Keywords: shape optimization, diameter, inradius, convex geometry, 2-cap bodies, Blaschke–Santaló diagram
Delyon, Alexandre 1; Henrot, Antoine 1; Privat, Yannick 2

1 Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239 54506 Vandœuvre-lès-Nancy Cedex (France)
2 IRMA, Université de Strasbourg, CNRS UMR 7501, Inria, 7 rue René Descartes, 67084 Strasbourg (France) Institut Universitaire de France (IUF)
     author = {Delyon, Alexandre and Henrot, Antoine and Privat, Yannick},
     title = {The missing $\protect (A,D,r)$ diagram},
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Delyon, Alexandre; Henrot, Antoine; Privat, Yannick. The missing $\protect (A,D,r)$ diagram. Annales de l'Institut Fourier, Online first, 52 p.

[1] Belloni, Marino; Oudet, Edouard The minimal gap between Λ 2 (Ω) and Λ (Ω) in a class of convex domains, J. Convex Anal., Volume 15 (2008) no. 3, pp. 507-521 | MR | Zbl

[2] Blaschke, Wilhelm Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Math. Ann., Volume 76 (1915), pp. 504-513 | Zbl

[3] Blaschke, Wilhelm Eine Frage über konvexe Körper, Jahresber. Dtsch. Math.-Ver., Volume 25 (1916), pp. 121-125 | Zbl

[4] Bonnesen, Tommy; Fenchel, Werner Theory of convex bodies, BCS Associates, Moscow (USA), 1987 (Translated from the German and edited by L. Boron, C. Christenson and B. Smith) | MR | Zbl

[5] Böröczky, Károly Jr.; Hernández Cifre, María A.; Salinas, Guillermo Optimizing area and perimeter of convex sets for fixed circumradius and inradius, Monatsh. Math., Volume 138 (2003) no. 2, pp. 95-110 | DOI | MR | Zbl

[6] Brandenberg, René; González Merino, Bernardo A complete 3-dimensional Blaschke–Santaló diagram, Math. Inequal. Appl., Volume 20 (2017) no. 2, pp. 301-348 | DOI | MR | Zbl

[7] Delyon, Alexandre; Henrot, Antoine; Privat, Yannick Nondispersal and density properties of infinite packings, SIAM J. Control Optim., Volume 57 (2019) no. 2, pp. 1467-1492 | DOI | MR | Zbl

[8] Henk, Martin; Tsintsifas, George A. Some inequalities for planar convex figures, Elem. Math., Volume 49 (1994) no. 3, pp. 120-125 | MR | Zbl

[9] Henrot, Antoine; Pierre, Michel Shape Variation and Optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | Zbl

[10] Hernández Cifre, María A. Is there a planar convex set with given width, diameter, and inradius?, Am. Math. Mon., Volume 107 (2000) no. 10, pp. 893-900 | DOI | MR | Zbl

[11] Hernández Cifre, María A. Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius, Arch. Math., Volume 79 (2002) no. 2, pp. 147-157 | DOI | MR | Zbl

[12] Hernández Cifre, María A.; Gomis, Segura S. The missing boundaries of the Santaló diagrams for the cases (d,ω,R) and (ω,R,r), Discrete Comput. Geom., Volume 23 (2000) no. 3, pp. 381-388 | DOI | MR | Zbl

[13] Hernández Cifre, María A.; Salinas, Guillermo Some optimization problems for planar convex figures, Rend. Circ. Mat. Palermo Suppl. (2002) no. 70, part I, pp. 395-405 IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures & Applications to Engineering Science”, Vol. I (Tropea, 2001) | MR | Zbl

[14] Jaglom, Isaak M.; Boltjanskiĭ, Vladimir G. Convex figures, Library of the Mathematical Circle, 4, Holt, Rinehart and Winston, 1961 (translated by Paul J. Kelly and Lewis F. Walton) | MR | Zbl

[15] Santaló, Luis A. On complete systems of inequalities between elements of a plane convex figure, Math. Notae, Volume 17 (1961), pp. 82-104 | MR | Zbl

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

[17] Scott, Paul R.; Awyong, Poh Wah Inequalities for convex sets, JIPAM, J. Inequal. Pure Appl. Math., Volume 1 (2000) no. 1, 6 | MR | Zbl

[18] Yang, Yunlong; Zhang, Deyan Two optimisation problems for convex bodies, Bull. Aust. Math. Soc., Volume 93 (2016) no. 1, pp. 137-145 | DOI | MR | Zbl

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