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]
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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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
@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, Tome 73 (2023) no. 2, pp. 635-694. doi : 10.5802/aif.3550. https://aif.centre-mersenne.org/articles/10.5802/aif.3550/

[1] Álvarez Paiva, J. C.; Thompson, A. C. Volumes on normed and Finsler spaces, A sampler of Riemann–Finsler geometry (Math. Sci. Res. Inst. Publ.), Volume 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1-48 | DOI | MR | Zbl

[2] Ambrosio, L.; Schmidt, Th. Compactness results for normal currents and the Plateau problem in dual Banach spaces, Proc. Lond. Math. Soc. (3), Volume 106 (2013) no. 5, pp. 1121-1142 | DOI | MR | Zbl

[3] Bouafia, Ph.; De Pauw, Th. Integral geometric measure in separable Banach space, Math. Ann., Volume 363 (2015) no. 1-2, pp. 269-304 | DOI | MR | Zbl

[4] Burago, D.; Ivanov, S. Gaussian images of surfaces and ellipticity of surface area functionals, Geom. Funct. Anal., Volume 14 (2004) no. 3, pp. 469-490 | DOI | MR | Zbl

[5] Burago, D.; Ivanov, S. Minimality of planes in normed spaces, Geom. Funct. Anal., Volume 22 (2012) no. 3, pp. 627-638 | DOI | MR | Zbl

[6] Busemann, H. Intrinsic area, Ann. of Math. (2), Volume 48 (1947), pp. 234-267 | DOI | MR | Zbl

[7] Busemann, H. A theorem on convex bodies of the Brunn-Minkowski type, Proc. Nat. Acad. Sci. U. S. A., Volume 35 (1949), pp. 27-31 | DOI | MR | Zbl

[8] De Pauw, Th. Size minimizing surfaces, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 1, pp. 37-101 | DOI | Numdam | MR | Zbl

[9] De Pauw, Th. Approximation by polyhedral G chains in Banach spaces, Z. Anal. Anwend., Volume 33 (2014) no. 3, pp. 311-334 | DOI | MR | Zbl

[10] De Pauw, Th.; Hardt, R. Size minimization and approximating problems, Calc. Var. Partial Differential Equations, Volume 17 (2003) no. 4, pp. 405-442 | DOI | MR | Zbl

[11] De Pauw, Th.; Hardt, R. Rectifiable and flat G chains in a metric space, Amer. J. Math., Volume 134 (2012) no. 1, pp. 1-69 | DOI | MR | Zbl

[12] De Pauw, Th.; Hardt, R. Some basic theorems on flat G chains, J. Math. Anal. Appl., Volume 418 (2014) no. 2, pp. 1047-1061 | DOI | MR | Zbl

[13] Federer, H. Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer-Verlag New York Inc., 1969, xiv+676 pages | MR | Zbl

[14] Federer, H. Flat chains with positive densities, Indiana Univ. Math. J., Volume 35 (1986) no. 2, pp. 413-424 | DOI | MR | Zbl

[15] Federer, H.; Fleming, W. H. Normal and integral currents, Ann. of Math. (2), Volume 72 (1960), pp. 458-520 | DOI | MR | Zbl

[16] Fleming, W. H. Flat chains over a finite coefficient group, Trans. Amer. Math. Soc., Volume 121 (1966), pp. 160-186 | DOI | MR | Zbl

[17] Kirchheim, B. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., Volume 121 (1994) no. 1, pp. 113-123 | DOI | MR | Zbl

[18] Morgan, F. Size-minimizing rectifiable currents, Invent. Math., Volume 96 (1989) no. 2, pp. 333-348 | DOI | MR | Zbl

[19] Parthasarathy, K. R. Probability measures on metric spaces, Probability and Mathematical Statistics, Academic Press, Inc., New York-London, 1967 no. 3, xi+276 pages | MR | Zbl

[20] Schneider, R. On the Busemann area in Minkowski spaces, Beiträge Algebra Geom., Volume 42 (2001) no. 1, pp. 263-273 | MR | Zbl

[21] Srivastava, S. M. A course on Borel sets, Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998, xvi+261 pages | DOI | MR | Zbl

[22] Thompson, A. C. Minkowski geometry, Encyclopedia of Mathematics and its Applications, 63, Cambridge University Press, Cambridge, 1996, xvi+346 pages | DOI | MR | Zbl

[23] White, B. Existence of least-energy configurations of immiscible fluids, J. Geom. Anal., Volume 6 (1996) no. 1, pp. 151-161 | DOI | MR | Zbl

[24] White, B. The deformation theorem for flat chains, Acta Math., Volume 183 (1999) no. 2, pp. 255-271 | DOI | MR | Zbl

[25] White, B. Rectifiability of flat chains, Ann. of Math. (2), Volume 150 (1999) no. 1, pp. 165-184 | DOI | MR | Zbl

[26] Xia, Q. Interior regularity of optimal transport paths, Calc. Var. Partial Differential Equations, Volume 20 (2004) no. 3, pp. 283-299 | DOI | MR | Zbl

Cité par Sources :