Stability of the relative Chebyshev projection in polyhedral spaces

I. G. Tsar'kov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 235-245

Résumé

The paper is concerned with structural and stability properties of the set of Chebyshev centers of a set. Given a nonempty bounded subset $M$ of a metric space $(X,\varrho)$, the quantity $\operatorname{diam} M =\sup_{x,y\in M}\varrho(x,y)$ is called the diameter of $M$, and $r_M:=r(M):=\inf\bigl\{a\geqslant 0, \ x\in X \mid M\subset B(x,a)\bigr\}$, the Chebyshev radius of $M$. A point $x_0\in X$ for which $M\subset B(x_0,r(M))$ is called a Chebyshev center of $M$. The concept of a Chebyshev center and related stability, existence and uniqueness problems are important in various branches of mathematics. We study the structure of the set of Chebyshev centers and the stability of the Chebyshev projection (the Chebyshev center map). In the space $X=C(Q)$, where $Q$ is a normal topological space, we describe the structure of the Chebyshev center of sets with a unique Chebyshev center. The Chebyshev projection is the mapping associating with a nonempty bounded set the set of all its Chebyshev centers. Given a nonempty bounded set $M$ of a space $X$ and a nonempty set $Y\subset X$, the relative Chebyshev radius is defined as $ r_Y(M)=\inf_{y\in Y} r(y,M)$, where $ ~r(x,M):=\inf\bigl\{r\ge 0\mid M\subset B(x,r)\bigr\}=\sup_{y\in M}\|x-y\|$. The set of relative Chebyshev centers is defined as $ ~\mathrm{Z}_Y(M):=\{y\in Y\mid r(y,M)=r_Y(M)\}$. The mapping $M\mapsto \mathrm{Z}_Y(M)$ is called the relative Chebyshev projection (with respect to the set $Y$). Stability properties of the relative Chebyshev projection in finite-dimensional polyhedral spaces are studied. In particular, in a finite-dimensional polyhedral space, the projection $\mathrm{Z}_Y(\,\cdot\,)$, where $Y$ is a subspace, is shown to be globally Lipschitz continuous.

Détail

Keywords: Chebyshev center, Chebyshev projection, stability.

Bibliographie

[1] Balashov M.V., Ivanov G.E., “Lipshitsevy parametrizatsii mnogoznachnykh otobrazhenii so slabo vypuklymi znacheniyami”, Izv. RAN. Ser. matematicheskaya, 71:6 (2007), 47–68 | DOI | MR | Zbl

[2] Balashov M.V., Repovŝ D., “On Pliś metric on the space of strictly convex compacta”, J. Convex Anal., 19:1 (2012), 171–183 | MR | Zbl

[3] Zikratova I.A., Shago F.N., Gurtov A.V., Ivaninskaya I.I., “Optimizatsiya zony pokrytiya seti sotovoi svyazi na osnove matematicheskogo programmirovaniya”, Nauch.-tekhn. vestnik inform. tekhnologii, mekhaniki i optiki, 15:2 (2015), 313–321 | MR

[4] Geniatulin K.A., Nosov V.I., “Primenenie metoda koordinatsionnykh kolets pri chastotno-territorialnom planirovanii sistemy sputnikovoi svyazi s zonalnym obsluzhivaniem”, Vestn. SibGUTI, 2014, no. 1, 35–45

[5] Bychkov I.V., Kazakov A.L., Lempert A.A., Bukharov D.S., Stolbov A.B., “Intellektnaya sistema upravleniya razvitiem transportno-logisticheskoi infrastrukturoi regiona”, Problemy upravleniya, 2014, no. 1, 27–35

[6] Guseinov Kh.G., Moiseev A.N., Ushakov V.N., “Ob approksimatsii oblastei dostizhimosti upravlyaemykh sistem”, Prikladnaya matematika i mekhanika, 62:2 (1998), 179–187 | MR

[7] Ivanov V.V., “Ob optimalnykh po tochnosti algoritmakh priblizhennogo resheniya operatornykh uravnenii I roda”, Zhurn. vychiclit. matematiki i mat. fiziki, 15:1 (1975), 3–11 | Zbl

[8] Ushakov V.N., Lebedev P.D., Lavrov N.G., “Algoritmy postroeniya optimalnykh upakovok v ellipsy”, Vestn. YuUrGU. Ser. Mat. modelirovanie i programmirovanie, 10:3 crossref{http://dx.doi.org/10.14529/mmp170306} (2017), 67–79 | Zbl

[9] Alimov A.R., Tsarkov I.G., “Svyaznost i solnechnost v zadachakh nailuchshego i pochti nailuchshego priblizheniya”, Uspekhi mat. nauk, 71:1(427) (2016), 3–81 | DOI | MR

[10] Vasileva A.A., “Zamknutye promezhutki v $C(T)$ i $L_\varphi(T)$ i ikh approksimativnye svoistva v normirovannykh prostranstvakh”, Mat. zametki, 73:1 crossref{http://dx.doi.org/10.4213/im496} (2003), 135–138 | DOI | MR | Zbl

[11] Vasileva A.A., “Zamknutye promezhutki v vektornoznachnykh funktsionalnykh prostranstvakh i ikh approksimativnye svoistva”, Izv. RAN. Ser. matematicheskaya, 68:4 (2004), 75–116 | DOI | MR | Zbl

[12] Garcia-Ferreira S., Ortiz-Castillo Y.F., Yamauchi T., “Insertion theorems for maps to linearly ordered topological spaces”, Topol. Appl., 188 (2015), 74–81 | DOI | MR | Zbl

[13] Franchetti C., Cheney E.W., “The embedding of proximinal sets”, J. Approx. Theor., 4 (1986), 213–225 | DOI | MR

[14] Levitin E.S., Teoriya vozmuschenii v matematicheskom programmirovanii i ee prilozheniya, Nauka, M., 1992, 359 pp.

[15] Polovinkin E.S.,Balashov M.V., Elementy vypuklogo i silno vypuklogo analiza, Fizmatlit, M., 2004, 416 pp.

[16] Druzhinin Yu.Yu., “O suschestvovanii lipshitsevoi vyborki iz chebyshëvskikh tsentrov”, Mat. sb., 204:5 (2013), 25–44 | DOI | MR | Zbl

[17] Cline A.K., “Lipschitz conditions on uniform approximation operators”, J. Approx. Theory, 8:2 (1973), 160–172 | DOI | MR | Zbl

[18] Berdyshev V.I., “Metricheskaya proektsiya na konechnomernye podprostranstva iz $C$ i $L$”, Mat. zametki, 18:4 (1975), 473–488 | DOI | MR | Zbl

[19] Bartelt M., “On Lipschitz conditions, strong unicity and a theorem of A. K. Cline”, J. Approx. Theory, 76:3 (1975), 245–250 | DOI | MR

[20] Finzel M., “Linear-approximation in $\ell^\infty_n$”, J. Approx. Theory, 76:3 (1994), 326–350 | DOI | MR | Zbl

[21] Li W., “Hoffman's theorem and metric projections in polyhedral spaces”, J. Approx. Theory, 75:1 (1993), 107–111 | DOI | MR | Zbl

[22] Finzel M., Li W., “Piecewise affine selections for piecewise polyhedral multifunctions and metric projections”, J. Conv. Anal., 7:1 (2000), 97–94 | MR