Tomographic characterizations of suns in three-dimensional spaces

A. R. Alimov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 45-55

Résumé

Recently A. R. Alimov and B. B. Bednov characterized the three-dimensional spaces in which any Chebyshev set is monotone path-connected. In particular, they showed that any Chebyshev set in a three-dimensional space with cylindrical norm is monotone path-connected. The author of the present paper obtained a similar result for closed sets with continuous (lower semicontinuous) metric projection. R. Aumann established that if the section of a compact subset $M$ of a finite-dimensional space by any hyperplane is acyclic, then $M$ is convex. A sun is considered as a possible generalization of a convex set — it is well known that any point not lying in a sun can be separated from it by an open support cone. In the present paper, we consider the problem of tomographic classification of suns in terms of approximative and geometric properties of their sections by tangent planes. We consider the case of three-dimensional spaces with cylindrical norm. In these spaces, we introduce the notion of a tangent plane, which generalizes the notion of a tangent direction to a sphere introduced by A. R. Alimov and E. V. Shchepin. The results obtained in the paper partially generalize and extend the mentioned studies. We give necessary and sufficient conditions for the monotone path-connectedness of approximatively defined sets in three-dimensional cylindrical spaces in terms of approximative and geometric properties of their sections by tangent planes.

Détail

Keywords: best approximation, Chebyshev set, sun, monotone path-connected set.

Bibliographie

[1] Alimov A.R., Tsarkov I.G., “Svyaznost i solnechnost v zadachakh nailuchshego i pochti nailuchshego priblizheniya”, Uspekhi mat. nauk, 71:1 (2016), 3–84 | DOI | MR | Zbl

[2] Alimov A.R., “Vypuklost i monotonnaya lineinaya svyaznost mnozhestv s nepreryvnoi metricheskoi proektsiei v trekhmernykh prostranstvakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 26:2 (2020), 29–46 | DOI

[3] Alimov A.R., “Geometricheskoe stroenie chebyshevskikh mnozhestv i solnts v trekhmernykh prostranstvakh s tsilindricheskoi normoi”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2020, no. 5, 26–32 | DOI | Zbl

[4] Alimov A.R., “Kharakterizatsiya mnozhestv s nepreryvnoi metricheskoi proektsiei v prostranstve $\ell^\infty_n$”, Mat. zametki, 108:3 (2020), 323–333 | DOI | MR | Zbl

[5] Alimov A.R., “Monotone path-connectedness of strict suns”, Lobachevskii J. Math., 43:3 (2022), 1267–1276 | DOI

[6] Alimov A.R., Bednov B.B., “Monotonnaya lineinaya svyaznost chebyshevskikh mnozhestv v trekhmernykh prostranstvakh”, Mat. sb., 212:5 (2021), 37–57 | DOI | MR | Zbl

[7] Alimov A.R., Shchepin E.V., “Convexity of suns in tangent directions”, J. Convex Analysis, 26:4 (2019), 1071–1076 | MR | Zbl

[8] Alimov A.R., Tsar'kov I. G., “Smoothness of subspace sections of the unit balls of $C (Q)$ and $L^1$”, J. Approx. Theory, 265 (2021), 105552 | DOI | MR | Zbl

[9] Aumann G., “On a topological characterization of compact convex point sets”, Ann. of Math., 37 (1936), 443–447 | DOI | MR

[10] Bednov B.B., “Konechnomernye prostranstva, v kotorykh klass chebyshevskikh mnozhestv sovpadaet s klassom zamknutykh i monotonno lineino svyaznykh mnozhestv”, Mat. zametki, 111:4 (2022), 483–493 | DOI | Zbl

[11] Bendit T., “Chebyshev subsets of a Hilbert space sphere”, J. Austral. Math. Soc., 107:3 (2019), 289–301 | DOI | MR | Zbl

[12] Bingham N.H., “The life, work, and legacy of P. L. Chebyshev”, Theory Probab. Appl., 66:4 (2022), 506–521 | DOI | MR | Zbl

[13] Boltyanski V., Martini H., Soltan P.S., Excursions into combinatorial geometry, Springer, Berlin, 1997, 427 pp. | MR | Zbl

[14] Brosowski B., Deutsch F., “On some geometric properties of suns”, J. Approx. Theory, 10:3 (1974), 245–267 | DOI | MR | Zbl

[15] Brown A.L., “Suns in normed linear spaces which are finite dimensional”, Math. Ann., 279:1 (1987), 87–101 | DOI | MR | Zbl

[16] Lebedev P.D., Uspenskii A.A., “Ob analiticheskom postroenii reshenii v odnom klasse zadach upravleniya po bystrodeistviyu s nevypuklym tselevym mnozhestvom”, Tr. In-ta matematiki i mekhaniki UrO RAN, 27:3 (2021), 128–140 | DOI | MR

[17] Montejano L., Shchepin E., “Topological tomography in convexity”, Bull. London Math. Soc., 34:3 (2002), 353–358 | DOI | MR | Zbl

[18] Nath T., “Differentiability of distance function and the proximinal condition implying convexity”, J. Analysis, 29:1 (2021), 247–261 | DOI | MR | Zbl

[19] Tsarkov I.G., “Slabo monotonnye mnozhestva i nepreryvnaya vyborka v nesimmetrichnykh prostranstvakh”, Mat. sb., 210:9 (2019), 129–155 | DOI | MR

[20] Tsarkov I.G., “Geometriya osobogo mnozhestva giperpoverkhnostei i uravnenie eikonala”, Mat. zametki, 108:3 (2020), 441–451 | DOI | MR | Zbl

[21] Tsarkov I.G., “Svoistva monotonno lineino svyaznykh mnozhestv”, Izv. RAN. Ser. matematicheskaya, 85:2 (2021), 142–171 | DOI | MR | Zbl

[22] Tsarkov I.G., “Svoistva monotonno svyaznykh mnozhestv”, Mat. zametki, 109:5 (2021), 781–792 | DOI | MR | Zbl

[23] Tsar'kov I.G., “Properties of suns in the spaces $L^ 1$ and $C (Q)$”, Russian J. Math. Physics, 28:3 (2021), 398–405 | DOI | MR | Zbl

[24] Tsarkov I.G., “Solnechnost i svyaznost mnozhestv v prostranstve $C[a,b]$ i konechnomernykh poliedralnykh prostranstvakh”, Mat. sb., 213:2 (2022), 149–166 | DOI | Zbl

[25] Shiryaev A. N., “K 200-letiyu so dnya rozhdeniya velikogo russkogo matematika P. L. Chebysheva”, Teoriya veroyatnosti i ee primenenie, 66:4 (2021), 625–635 | DOI