The convex hull and the Carathéodory number of a set in terms of the metric projection operator

K. S. Shklyaev

Sbornik. Mathematics, Tome 213 (2022) no. 10, pp. 1470-1486

Résumé

We prove that each point of the convex hull of a compact set $M$ in a smooth Banach space $X$ can be approximated arbitrarily well by convex combinations of best approximants from $M$ to $x$ (values of the metric projection operator $P_M(x)$), where $x \in X$. As a corollary, we show that the Carathéodory number of a compact set $M \subset X$ with at most $k$-valued metric projection $P_M$ is majorized by $k$, that is, each point in the convex hull of $M$ lies in the convex hull of at most $k$ points of $M$. Bibliography: 26 titles.

Détail

Keywords: metric projection, convex hull, Banach space, smoothness, Minkowski functional
Mots-clés : Carathéodory number.

Bibliographie

[1] N. V. Efimov and S. B. Stechkin, “Some properties of Chebyshev sets”, Dokl. Akad. Nauk SSSR, 118:1 (1958), 17–19 (Russian) | MR | Zbl

[2] V. L. Klee, Jr., “A characterization of convex sets”, Amer. Math. Monthly, 56:4 (1949), 247–249 | DOI | MR | Zbl

[3] V. L. Klee, “Convex bodies and periodic homeomorphisms in Hilbert space”, Trans. Amer. Maths. Soc., 74 (1953), 10–43 | DOI | MR | Zbl

[4] V. I. Berdyshev, “On Chebyshev sets”, Dokl. Akad. Nauk Az. SSR, 22:9 (1966), 3–5 (Russian) | MR | Zbl

[5] A. Brøndsted, “Convex sets and Chebyshev sets. II”, Math. Scand., 18 (1966), 5–15 | DOI | MR | Zbl

[6] A. L. Brown, “Chebyshev sets and the shapes of convex bodies”, Methods of functional analysis in approximation theory (Bombay 1985), Internat. Schriftenreihe Numer. Math., 76, Birkhäuser, Basel, 1986, 97–121 | MR | Zbl

[7] A. L. Brown, “Chebyshev sets and facial systems of convex sets in finite-dimensional spaces”, Proc. London Math. Soc. (3), 41:2 (1980), 297–339 | DOI | MR | Zbl

[8] L. P. Vlasov, “Approximative properties of sets in normed linear spaces”, Uspekhi Mat. Nauk, 28:6(174) (1973), 3–66 ; English transl. in Russian Math. Surveys, 28:6 (1973), 1–66 | MR | Zbl | DOI

[9] V. S. Balaganskii and L. P. Vlasov, “The problem of convexity of Chebyshev sets”, Uspekhi Mat. Nauk, 51:6(312) (1996), 125–188 ; English transl. in Russian Math. Surveys, 51:6 (1996), 1127–1190 | DOI | MR | Zbl | DOI

[10] I. G. Tsar'kov, “Bounded Chebyshev sets in finite-dimensional Banach spaces”, Mat. Zametki, 36:1 (1984), 73–87 ; English transl. in Math. Notes, 36:1 (1984), 530–537 | MR | Zbl | DOI

[11] I. G. Tsar'kov, “Compact and weakly compact Tchebysheff sets in normed linear spaces”, Proceeding of the All-Union school on the theory of functions, Tr. Mat. Inst. Steklov., 189, Nauka, Moscow, 1989, 169–184 ; English transl. in Proc. Steklov Inst. Math., 189 (1990), 199–215 | MR | Zbl

[12] A. R. Alimov, “Is every Chebyshev set convex?”, Mat. Prosveshchenie, Ser. 3, 2, Moscow Center for Continuous Mathematical Education, Moscow, 1998, 155–172 (Russian)

[13] A. R. Alimov, “On the structure of the complements of Chebyshev sets”, Funktsional. Anal. i Prilozhen., 35:3 (2001), 19–27 ; English transl. in Funct. Anal. Appl., 35:3 (2001), 176–182 | DOI | MR | Zbl | DOI

[14] L. N. H. Bunt, Bijdrage tot de theorie der convexe puntverzamelingen, Proefschrifft Groningen, Noord-Hollandsche Uitgevers Maatschappij, Amsterdam, 1934, 108 pp. | Zbl

[15] H. Mann, “Untersuchungen über Wabenzellen bei allgemeiner Minkowskischer Metrik”, Monatsh. Math. Phys., 42:1 (1935), 417–424 | DOI | MR | Zbl

[16] T. Motzkin, “Sur quelques propriétés caractéristiques des ensembles convexes”, Atti Accad. Naz. Lincei Rend. (6), 21 (1935), 562–567 | Zbl

[17] T. Motzkin, “Sur quelques propriétés caractéristiques des ensembles bornés non convexes”, Atti Accad. Naz. Lincei Rend. (6), 21 (1935), 773–779 | Zbl

[18] C. Carathéodory, “Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen”, Rend. Circ. Mat. Palermo, 32 (1911), 193–217 | DOI | Zbl

[19] W. Fenchel, “Über Krümmung und Windung geschlossener Raumkurven”, Math. Ann., 101:1 (1929), 238–252 | DOI | MR | Zbl

[20] I. Bárány and R. Karasev, “Notes about the Carathéodory number”, Discrete Comput. Geom., 48:3 (2012), 783–792 | DOI | MR | Zbl

[21] A. A. Flerov, “Sets with at most two-valued metric projection on a normed plane”, Mat. Zametki, 101:2 (2017), 286–301 ; English transl. in Math. Notes, 101:2 (2017), 352–364 | DOI | MR | Zbl | DOI

[22] O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and V. M. Kharlamov, Elementary topology. Problem textbook, 3d ed., Moscow Center for Continuous Mathematical Education, Moscow, 2010, 446 pp.; English transl., Amer. Math. Soc., Providence, RI, 2008, xx+400 pp. | DOI | MR | Zbl

[23] A. Cellina, “Approximation of set valued functions and fixed point theorems”, Ann. Mat. Pura Appl. (4), 82 (1969), 17–24 | DOI | MR | Zbl

[24] M. L. Gromov, “On simplexes inscribed in a hypersurface”, Mat. Zametki, 5:1 (1969), 81–89 ; English transl. in Math. Notes, 5:1 (1969), 52–56 | MR | Zbl | DOI

[25] A. Brøndsted, “Convex sets and Chebyshev sets”, Math. Scand., 17 (1965), 5–16 | DOI | MR | Zbl

[26] Ş. Cobzaş, Functional analysis in asymmetric normed spaces, Front. Math., Birkhäuser/Springer Basel AG, Basel, 2013, x+219 pp. | DOI | MR | Zbl