The Hausdorff lower semicontinuous envelope of the length in the plane

Annali della Scuola Normale Superiore di Pisa- Classe di Scienze 1 (1):33-71 (2002)
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Abstract

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

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