Abstract

We look for the minimizers of the functional Jλ(Ω)=λ∣Ω∣−P(Ω)\mathrm{J}_\lambda(\Omega) = \lambda|\Omega| - P(\Omega) among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter λ, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.

Author information

Contact details are reproduced from the original publication and may be historical.

Chiara Bianchini

Institut Elie Cartan, Université Henri Poincaré, Blvd des Aiguillettes, B.P. 70239, 54506 Vandoeuvre-les-Nancy, France

chiara.bianchini@iecn.u-nancy.fr

Antoine Henrot

Institut Elie Cartan, Université Henri Poincaré, Blvd des Aiguillettes, B.P. 70239, 54506 Vandoeuvre-les-Nancy, France

antoine.henrot@iecn.u-nancy.fr

Keywords

  • Convex geometry
  • shape optimization
  • isoperimetric inequalities
  • length
  • area

Mathematics Subject Classification

52A10, 52A38, 52A40, 49Q10

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Suggested citation

C. Bianchini, A. Henrot. “Optimal Sets for a Class of Minimization Problems with Convex Constraints.” Journal of Convex Analysis 19 (2012), No. 3, 725–758. https://doi.org/10.68381/jca19039

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.