Criteria for Convexity of Closed Sets in Banach Spaces

Yu. S. Ledyaev

Informatics and Automation, Optimal control and differential equations, Tome 304 (2019), pp. 205-220.

Résumé

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.

Bibliographie

[1] Ben-El-Mechaiekh H., Kryszewski W., “Equilibria of set-valued maps on nonconvex domains”, Trans. Amer. Math. Soc., 349:10 (1997), 4159–4179 | DOI | MR | Zbl

[2] N. A. Bobylev, S. V. Emel'yanov, and S. K. Korovin, “Convexity of images of convex sets under smooth maps”, Comput. Math. Model., 15:3 (2004), 213–222 | DOI | MR | Zbl

[3] Borwein J.M., Giles J.R., “The proximal normal formula in Banach space”, Trans. Amer. Math. Soc., 302:1 (1987), 371–381 | DOI | MR | Zbl

[4] Borwein J.M., Strojwas H.M., “Proximal analysis and boundaries of closed sets in Banach space. II: Applications”, Can. J. Math., 39:2 (1987), 428–472 | DOI | MR | Zbl

[5] Clarke F.H., Optimization and nonsmooth analysis, Can. Math. Soc. Ser. Monogr. Adv. Texts, J. Wiley Sons, New York, 1983 | MR | Zbl

[6] Clarke F.H., Methods of dynamic and nonsmooth optimization, CBMS–NSF Reg. Conf. Ser. Appl. Math., 57, SIAM, Philadelphia, PA, 1989 | MR | Zbl

[7] Clarke F.H., Necessary conditions in dynamic optimization, Mem. AMS, 173, no. 816, Amer. Math. Soc., Providence, RI, 2005 | MR

[8] Clarke F.H., Ledyaev Yu.S., Stern R.J., “Fixed points and equilibria in nonconvex sets”, Nonlinear Anal. Theory Methods Appl., 25:2 (1995), 145–161 | DOI | MR | Zbl

[9] Clarke F.H., Ledyaev Yu.S., Stern R.J., “Fixed point theory via nonsmooth analysis”, Recent developments in optimization theory and nonlinear analysis, AMS/IMU Spec. Sess. (Jerusalem, 1995), Contemp. Math., 204, Amer. Math. Soc., Providence, RI, 1997, 93–106 | DOI | MR | Zbl

[10] Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and control theory, Grad. Texts Math., 178, Springer, New York, 1998 | MR | Zbl

[11] S. V. Emel'yanov, S. K. Korovin, and N. A. Bobylev, “On the convexity of the images of convex sets under smooth mappings”, Dokl. Math., 66:1 (2002), 55–57 | MR | Zbl

[12] Klee V.L., \textup {Jr., “} Convex sets in linear spaces”, Duke Math. J., 18 (1951), 443–466 | DOI | MR | Zbl

[13] Lebourg G., “Review of the paper “The convexity principle and its applications” by B.T. Polyak”, Math. Rev., 2004 | MR

[14] Ledyaev Yu.S., Zhu Q.J., “Implicit multifunction theorems”, Set-Valued Anal., 7:3 (1999), 209–238 | DOI | MR | Zbl

[15] Loewen P.D., “The proximal normal formula in Hilbert space”, Nonlinear Anal. Theory Methods Appl., 11:9 (1987), 979–995 | DOI | MR | Zbl

[16] Polyak B.T., “Convexity of nonlinear image of a small ball with applications to optimization”, Set-Valued Anal., 9:1-2 (2001), 159–168 | DOI | MR | Zbl

[17] Polyak B.T., “The convexity principle and its applications”, Bull. Braz. Math. Soc. (N.S.), 34:1 (2003), 59–75 | DOI | MR | Zbl

[18] G. Reißig, “Convexity of reachable sets of nonlinear ordinary differential equations”, Autom. Remote Control, 68:9 (2007), 1527–1543 | DOI | MR | Zbl

[19] Treiman J.S., “Characterization of Clarke's tangent and normal cones in finite and infinite dimensions”, Nonlinear Anal. Theory Methods Appl., 7:7 (1983), 771–783 | DOI | MR | Zbl

[20] Uderzo A., “On the Polyak convexity principle and its application to variational analysis”, Nonlinear Anal. Theory Methods Appl., 91 (2013), 60–71 | DOI | MR | Zbl

[21] Uderzo A., “Convexity of the images of small balls through nonconvex multifunctions”, Nonlinear Anal. Theory Methods Appl., 128 (2015), 348–364 | DOI | MR | Zbl

[22] S. A. Vakhrameev, “A note on convexity in smooth nonlinear systems”, J. Math. Sci., 100:5 (2000), 2470–2490 | DOI | MR