We introduce the notion of density contractor of dimension in a finite dimensional normed space . If , this includes the area contracting projectors on hyperplanes whose existence was established by H. Busemann. If , density contractors are an ersatz for such projectors and their existence, established here, follows from works by D. Burago and S. Ivanov. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable chains, regardless of the group of coefficients . This is obtained as a consequence of the lower semicontinuity of the dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.
Nous introduisons la notion de contracteur de densité de dimension , dans un espace normé de dimension finie. Lorsque , celle-ci inclut les projecteurs contractants sur les hyperplans, dont l’existence a été établie par H. Busemann. Lorsque , les contracteurs de densité constituent un ersatz à ces projecteurs, et leur existence, établie ci-dessous, découle de travaux de D. Burago et S. Ivanov. En présence de contracteurs de densité, le problème de Plateau correspondant admet une solution parmi les chaînes rectifiables, indépendamment du groupe de coefficients . Ceci est une conséquence de la semi-continuité inférieure de la masse de Hausdorff, dont nous proposons deux démonstrations. L’une d’elles repose sur un nouveau type de mesure intégrale géométrique.
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Accepted:
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Keywords: Hausdorff measure, integral geometry, rectifiable chains, Plateau problem.
Mot clés : Mesure de Hausdorff, géométrie intégrale, chaînes rectifiables, problème de Plateau.
De Pauw, Thierry 1; Vasilyev, Ioann 2
@article{AIF_2023__73_2_635_0, author = {De Pauw, Thierry and Vasilyev, Ioann}, title = {On the existence of mass minimizing rectifiable $G$ chains in finite dimensional normed spaces}, journal = {Annales de l'Institut Fourier}, pages = {635--694}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {2}, year = {2023}, doi = {10.5802/aif.3550}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3550/} }
TY - JOUR AU - De Pauw, Thierry AU - Vasilyev, Ioann TI - On the existence of mass minimizing rectifiable $G$ chains in finite dimensional normed spaces JO - Annales de l'Institut Fourier PY - 2023 SP - 635 EP - 694 VL - 73 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3550/ DO - 10.5802/aif.3550 LA - en ID - AIF_2023__73_2_635_0 ER -
%0 Journal Article %A De Pauw, Thierry %A Vasilyev, Ioann %T On the existence of mass minimizing rectifiable $G$ chains in finite dimensional normed spaces %J Annales de l'Institut Fourier %D 2023 %P 635-694 %V 73 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3550/ %R 10.5802/aif.3550 %G en %F AIF_2023__73_2_635_0
De Pauw, Thierry; Vasilyev, Ioann. On the existence of mass minimizing rectifiable $G$ chains in finite dimensional normed spaces. Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 635-694. doi : 10.5802/aif.3550. https://aif.centre-mersenne.org/articles/10.5802/aif.3550/
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