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Journal of Convex Analysis 12 (2005), No. 1, 239--253
Copyright Heldermann Verlag 2005

Γ Convergence of Hausdorff Measures

Giuseppe Buttazzo
Dip. di Matematica, Universitŗ di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

Benjamin Schweizer
Inst. Angewandte Mathematik, Universitšt Heidelberg, INF 294, 69120 Heidelberg, Germany


\def\H1{\mathcal{H}^1} We study the dependence of the Hausdorff measure $\H1_d$ on the distance $d$. We show that the uniform convergence of $d_j$ to $d$ is equivalent to the $\Gamma$ convergence of $\H1_{d_j}$ to $\H1_d$ with respect to the Hausdorff convergence on compact connected subsets. We also consider the case when distances are replaced by semi-distances.

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