# 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

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topLemenant, 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 -

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