no. 2
On the existence of mass minimizing rectifiable G chains in finite dimensional normed spaces
[Sur l’existence de G chaînes rectifiables minimisant la masse dans des espaces normés de dimension finie]
De Pauw, Thierry1 ; Vasilyev, Ioann2
1 Université Paris Cité and Sorbonne Université CNRS IMJ-PRG F-75013 Paris (France)
2 St. Petersburg Department of Steklov Mathematical Institute Russian Academy of Sciences (PDMI RAS) (Russia)
Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 635-694.

Nous introduisons la notion de contracteur de densité de dimension m, dans un espace normé X de dimension finie. Lorsque m+1=dimX, celle-ci inclut les projecteurs contractants sur les hyperplans, dont l’existence a été établie par H. Busemann. Lorsque m=2, 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 G chaînes rectifiables, indépendamment du groupe de coefficients G. 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.

We introduce the notion of density contractor of dimension m in a finite dimensional normed space X. If m+1=dimX, this includes the area contracting projectors on hyperplanes whose existence was established by H. Busemann. If m=2, 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 G chains, regardless of the group of coefficients G. This is obtained as a consequence of the lower semicontinuity of the m dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.

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DOI : 10.5802/aif.3550
Classification : 49Q15, 49Q20, 52A21, 28A75, 52A38, 52A40, 49J45
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

1 Université Paris Cité and Sorbonne Université CNRS IMJ-PRG F-75013 Paris (France)
2 St. Petersburg Department of Steklov Mathematical Institute Russian Academy of Sciences (PDMI RAS) (Russia)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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De Pauw, Thierry; Vasilyev, Ioann. On the existence of mass minimizing rectifiable $G$ chains in finite dimensional normed spaces. Annales de l'Institut Fourier, Tome 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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