Hyperspace of Max-Plus Convex Compact Sets

L. E. Bazilevich

Matematičeskie zametki, Tome 84 (2008) no. 5, pp. 658-666.

Résumé

The hyperspace $\operatorname{mpcc}(\mathbb R^n)$ of max-plus convex compact subsets of $\mathbb R^n$, $n\ge2$, is considered. The main result is as follows: this hyperspace is a contractible manifold modeled on the Hilbert cube $Q$. It is also shown that the projection mapping $\operatorname{mpcc}(\mathbb R^n) \to\operatorname{mpcc}(\mathbb R^m)$, $n\ge m$, is open. Moreover, it is proved that the hyperspace $\operatorname{mpcc}(I^{\omega_1})$ of the Tikhonov [Tychonoff] cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$.

Détail

Keywords: compact convex set, max-plus convexity, contractible manifold, topological linear space, Hilbert cube, Tikhonov cube, separable metric space.
Mots-clés : retract

Bibliographie

[1] S. B. Nadler, Jr., J. Quinn, N. M. Stavrokas, “Hyperspace of compact convex sets”, Pacific J. Math., 83:2 (1979), 441–462 | MR | Zbl

[2] D. W. Curtis, R. M. Schori, “Hyperspaces of Peano continua are Hilbert cubes”, Fund. Math., 101:1 (1978), 19–38 | MR | Zbl

[3] L. E. Bazylevych, “On the hyperspace of strictly convex bodies”, Matem. studiï, Pratsi Lviv. matem. tov., 2, Lviv. matem. tov., Lviv, 1993, 83–86 | MR | Zbl

[4] Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, eds. G. L. Litvinov, V. P. Maslov, Amer. Math. Soc., Providence, RI, 2005 | MR | Zbl

[5] S. Gaubert, R. Katz, “The Minkowski theorem for max-plus convex sets”, Linear Algebra Appl., 421:2–3 (2007), 356–369 | DOI | MR | Zbl

[6] S. Gaubert, R. Katz, “Max-plus convex geometry”, Relations and Kleene Algebra in Computer Science, Lecture Notes in Comput. Sci., 4136, ed. R. A. Schmidt, Spriger, Berlin, 2006, 192–206 | MR | Zbl

[7] G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Idempotentnyi funktsionalnyi analiz. Algebraicheskii podkhod”, Matem. zametki., 69:5 (2001), 758–797 | MR | Zbl

[8] C. D. Horvath, “Contractibility and generalized convexity”, J. Math. Anal. Appl., 156:2 (1991), 341–357 | DOI | MR | Zbl

[9] C. D. Horvath, “Extension and selection theorems in topological spaces with a generalized convexity structure”, Ann. Fac. Sci. Toulouse Math. (6), 2:2 (1993), 253–269 | MR | Zbl

[10] M. Zarichnii, S. Ivanov,, “Giperprostori kompaktnikh opuklikh mnozhin tikhonovskogo kuba”, Ukr. matem. zhurn., 53:5 (2001), 698–701 | MR | Zbl

[11] J. van Mill, Infinite-Dimensional Topology. Prerequisites and Introduction, North-Holland Math. Library, 43, North-Holland Publ., Amsterdam, 1989 | MR | Zbl

[12] H. Toruńczyk, “On $\mathrm{CE}$-images of the Hilbert cube and characterization of $Q$-manifolds”, Fund. Math., 106:1 (1980), 31–40 | MR | Zbl

[13] E. B. Schepin, “Funktory i neschetnye stepeni kompaktov”, UMN, 36:3 (1981), 3–62 | MR | Zbl

[14] H. Toruńczyk, J. West, Fibrations and bundles with Hilbert cube manifold fibers, Mem. Amer. Math. Soc., 80, no. 406, Amer. Math. Soc., Providence, RI, 1989 | MR

[15] E. Michael, “Continuous selections. I”, Ann. of Math. (2), 63:2 (1956), 361–382 | DOI | MR | Zbl

[16] T. A. Chapman, “On some applications of infinite-dimensional manifolds to the theory of shape”, Fund. Math., 76:3 (1972), 181–193 | MR | Zbl

[17] C. Sirota, “Spektralnoe predstavlenie prostranstv zamknutykh podmnozhestv bikompaktov”, Dokl. AN SSSR., 181 (1968), 1069–1072 | MR | Zbl

[18] L. Montejano, “The hyperspace of compact convex subsets of an open subset of $R^n$”, Bull. Polish Acad. Sci. Math., 35:11–12 (1987), 793–799 | MR | Zbl

[19] C. Bessaga, A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, 58, PWN – Polish Sci. Publ., Warsaw, 1975 | MR | Zbl