Plongements quasiisométriques du groupe de Heisenberg dans $L^p$, d’après Cheeger, Kleiner, Lee, Naor

Pansu, Pierre

Séminaire de théorie spectrale et géométrie, Tome 25 (2006-2007), pp. 159-176.

Bref survol du théorème de non-plongement de J. Cheeger et B. Kleiner pour le groupe d’Heisenberg dans $L^1$.

This is a short survey of Cheeger and Kleiner’s nonembeddability theorem for Heisenberg group into $L^1$.

Classification : 20F65, 46B03, 46B22, 49Q15, 68Q17, 68W25
Mot clés : lipschitzien, plongement, espace de Banach, périmètre, groupe d’Heisenberg, algorithme
Mots clés : Lipschitz, embedding, Banach space, perimeter, Heisenberg group, algorithm

[1] Ambrosio, Luigi Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math., Volume 159 (2001) no. 1, pp. 51-67 | MR | Zbl

[2] Arora, Sanjeev; Hazan, Elad; Kale, Satyen; Society, IEEE Computer $O\left(\sqrt{logn}\right)$ approximation to SPARSEST CUT in $O\left(n^2\right)$ time, 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), Los Alamitos, CA (2004)

[3] Arora, Sanjeev; Lee, James R.; Naor, Assaf Euclidean distortion and the sparsest cut, J. Amer. Math. Soc., Volume 21 (2008) no. 1, p. 1-21 (electronic) | MR | Zbl

[4] Assouad, Patrice Espaces métriques, plongements, facteurs, U.E.R. Mathématique, Université Paris XI, Orsay, 1977 (Thèse de doctorat, Publications Mathématiques d’Orsay, No. 223-7769) | MR | Zbl

[5] Assouad, Patrice Plongements lipschitziens dans ${\mathbf{R}}^n$, Bull. Soc. Math. France, Volume 111 (1983) no. 4, pp. 429-448 | Numdam | MR | Zbl

[6] Bourgain, J. On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math., Volume 52 (1985) no. 1-2, pp. 46-52 | MR | Zbl

[7] Bourgain, J. The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., Volume 56 (1986) no. 2, pp. 222-230 | MR | Zbl

[8] Bretagnolle, Jean; Dacunha-Castelle, Didier; Krivine, Jean-Louis Lois stables et espaces $L^p$, Ann. Inst. H. Poincaré Sect. B (N.S.), Volume 2 (1965/1966), pp. 231-259 | Numdam | MR | Zbl

[9] Cheeger, Jeff; Kleiner, Bruce Differentiating maps into $L^1$, and the geometry of BV functions, arXiv :math/0611954

[10] Cheeger, Jeff; Kleiner, Bruce On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, Inspired by S. S. Chern (Nankai Tracts Math.), Volume 11, World Sci. Publ., Hackensack, NJ, 2006, pp. 129-152 | MR

[11] De Giorgi, Ennio Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat., Volume 4 (1955), pp. 95-113 | MR | Zbl

[12] Deza, Michel Marie; Laurent, Monique Geometry of cuts and metrics, Algorithms and Combinatorics, 15, Springer-Verlag, Berlin, 1997 | MR | Zbl

[13] Enflo, Per On the nonexistence of uniform homeomorphisms between $L_p$-spaces, Ark. Mat., Volume 8 (1969), p. 103-105 (1969) | MR | Zbl

[14] Franchi, Bruno; Serapioni, Raul; Serra Cassano, Francesco On the structure of finite perimeter sets in step 2 Carnot groups, J. Geom. Anal., Volume 13 (2003) no. 3, pp. 421-466 | MR | Zbl

[15] Goemans, Michel X. Semidefinite programming in combinatorial optimization, Math. Programming, Volume 79 (1997) no. 1-3, Ser. B, pp. 143-161 Lectures on mathematical programming (ismp97) (Lausanne, 1997) | MR | Zbl

[16] Gromov, M. Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Math. Soc. Lecture Note Ser.), Volume 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295 | MR

[17] Gromov, M. Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | MR | Zbl

[18] Heinonen, Juha; Koskela, Pekka From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. U.S.A., Volume 93 (1996) no. 2, pp. 554-556 | MR | Zbl

[19] Kadecʼ, M. Ĭ. On the connection between weak and strong convergence, Dopovidi Akad. Nauk Ukraïn. RSR, Volume 1959 (1959), pp. 949-952 | MR | Zbl

[20] Kakutani, Shizuo Mean ergodic theorem in abstract $\left(L\right)$-spaces, Proc. Imp. Acad., Tokyo, Volume 15 (1939), pp. 121-123 | MR | Zbl

[21] Khot, Subhash; Vishnoi, Nisheeth K.; Society, IEEE Computer The Unique Games Conjecture, Integrality Gap for Cut Problems and the Embeddability of Negative Type Metrics into $l_1$., 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), Los Alamitos, CA (2005), pp. 53-62

[22] Klee, V. Mappings into normed linear spaces, Fund. Math., Volume 49 (1960/1961), pp. 25-34 | MR | Zbl

[23] Lee, James; Naor, Assaf; Society, IEEE Computer $L_p$ metrics on the Heisenberg group and the Goemans-Linial conjecture, 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), Los Alamitos, CA (2005), p. 99-108, Preprint (2006)

[24] Linial, Nathan; Matousek, J. Squared ${\ell }_2$ metrics into ${\ell }_1$, Open Problems, Haifa, Workshop on Discrete Metric Spaces and their Algorithmic Applications (2002)

[25] Linial, Nathan; London, Eran; Rabinovich, Yuri The geometry of graphs and some of its algorithmic applications, Combinatorica, Volume 15 (1995) no. 2, pp. 215-245 | MR | Zbl

[26] Pauls, Scott D. The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom., Volume 9 (2001) no. 5, pp. 951-982 | MR | Zbl

[27] Semmes, Stephen On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_{\infty }$-weights, Rev. Mat. Iberoamericana, Volume 12 (1996) no. 2, pp. 337-410 | MR | Zbl

[28] Tessera, Romain Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces Preprint (2006)

[29] Yu, Guoliang The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., Volume 139 (2000) no. 1, pp. 201-240 | MR | Zbl