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We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.
@article{ASNSP_2002_5_1_1_33_0,
author = {Cerf, Rapha\"el},
title = {The {Hausdorff} lower semicontinuous envelope of the length in the plane},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {33--71},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {1},
year = {2002},
mrnumber = {1994801},
zbl = {1047.28001},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2002_5_1_1_33_0/}
}
TY - JOUR AU - Cerf, Raphaël TI - The Hausdorff lower semicontinuous envelope of the length in the plane JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2002 SP - 33 EP - 71 VL - 1 IS - 1 PB - Scuola normale superiore UR - http://geodesic.mathdoc.fr/item/ASNSP_2002_5_1_1_33_0/ LA - en ID - ASNSP_2002_5_1_1_33_0 ER -
%0 Journal Article %A Cerf, Raphaël %T The Hausdorff lower semicontinuous envelope of the length in the plane %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2002 %P 33-71 %V 1 %N 1 %I Scuola normale superiore %U http://geodesic.mathdoc.fr/item/ASNSP_2002_5_1_1_33_0/ %G en %F ASNSP_2002_5_1_1_33_0
Cerf, Raphaël. The Hausdorff lower semicontinuous envelope of the length in the plane. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 33-71. http://geodesic.mathdoc.fr/item/ASNSP_2002_5_1_1_33_0/