On convex sets that minimize the average distance

Antoine Lemenant; Edoardo Mainini

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 1049-1072
  • ISSN: 1292-8119

Abstract

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In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize ∫ Ω ( , K ) d + λ 1 Vol ( K ) + λ 2 Per ( K ) ∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from Tilli [J. Convex Anal. 17 (2010) 583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.

How to cite

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Lemenant, Antoine, and Mainini, Edoardo. "On convex sets that minimize the average distance." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1049-1072. <http://eudml.org/doc/272890>.

@article{Lemenant2012,
abstract = {In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize\begin\{equation*\} \int \_\{\Omega \} (,K) \,\{\rm d\}+ \lambda \_1 \{\rm Vol\}(K)+\lambda \_2 \{\rm Per\}(K) \end\{equation*\}∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from Tilli [J. Convex Anal. 17 (2010) 583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.},
author = {Lemenant, Antoine, Mainini, Edoardo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {shape optimization; distance functional; optimality conditions; convex analysis; second order variation; gamma-convergence; -convergence},
language = {eng},
number = {4},
pages = {1049-1072},
publisher = {EDP-Sciences},
title = {On convex sets that minimize the average distance},
url = {http://eudml.org/doc/272890},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Lemenant, Antoine
AU - Mainini, Edoardo
TI - On convex sets that minimize the average distance
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 4
SP - 1049
EP - 1072
AB - In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize\begin{equation*} \int _{\Omega } (,K) \,{\rm d}+ \lambda _1 {\rm Vol}(K)+\lambda _2 {\rm Per}(K) \end{equation*}∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from Tilli [J. Convex Anal. 17 (2010) 583–595] we can prove that any arbitrary convex set can be a minimizer when both perimeter and volume constraints are considered.
LA - eng
KW - shape optimization; distance functional; optimality conditions; convex analysis; second order variation; gamma-convergence; -convergence
UR - http://eudml.org/doc/272890
ER -

References

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