Sangeetha, M. Veena; Veeramani, P. Uniform rotundity with respect to finite-dimensional subspaces. (English) Zbl 1411.46013 J. Convex Anal. 25, No. 4, 1223-1252 (2018). The authors modify F.Sullivan’s modulus of \(k\)-uniform rotundity by introducing a modulus of uniform rotundity with respect to a finite-dimensional subspace \(Y\) of a normed space \(X\). To be specific, if \(Y\) is a \(k\)-dimensional subspace of a normed space \(X\) and \(\varepsilon>0\), let \(\delta_X(\varepsilon, Y)\) denote the infimum of the values \(1 - \frac{\|x_1 + \cdots + x_{k+1}\|}{k+1}\) where \(x_1, \dots, x_{k+1}\) are in the unit sphere of \(X\), the \(k\)-dimensional volume enclosed by the vectors \(\{x_1, \dots, x_{k+1}\}\) is at least \(\varepsilon\), and \({\text{span}} \{x_1- x_{k+1}, \dots, x_k-x_{k+1}\}=Y\). \(X\) is defined to be uniformly rotund with respect to \(Y\) if \(\delta_X(\varepsilon, Y)>0\) for every \(\varepsilon>0\); and, if \(X\) is uniformly rotund with respect to every \(k\)-dimensional subspace of \(X\), \(X\) is said to be URE\(_k\). The authors note that \(X\) is URE\(_1\) exactly, when \(X\) is uniformly rotund in every direction.The main purpose of the article is to generalize A.L.Garkavi’s result that a normed space is uniformly rotund in every direction if and only if the Chebyshev center of any nonempty bounded set contains at most one point. Define the dimension of a nonempty convex subset \(C\) to be the dimension of the linear space \(\text{span} (C-x)\) for any \(x\in C\). The authors prove that, if \(k\in\mathbb{N}\), a normed space \(X\) is URE\(_k\) if and only if the Chebyshev center of any nonempty bounded subset of \(X\) is either empty or of dimension at most \(k-1\). The authors also prove that normed spaces that are URE\(_k\) for some \(k\in\mathbb{N}\) have normal structure and provide sufficient conditions for the product of spaces that are uniformly rotund with respect to a finite-dimensional space \(Y\) to be uniformly rotund with respect to \(Y\). Reviewer: Barry Turett (Rochester) Cited in 8 Documents MSC: 46B20 Geometry and structure of normed linear spaces 47H10 Fixed-point theorems Keywords:uniform rotundity with respect to finite-dimensional subspaces; \(k\)-uniform rotundity; multi-dimensional volumes; Chebyshev centers; asymptotic centers; normal structure PDFBibTeX XMLCite \textit{M. V. Sangeetha} and \textit{P. Veeramani}, J. Convex Anal. 25, No. 4, 1223--1252 (2018; Zbl 1411.46013) Full Text: Link References: [1] D. Amir and Z. Ziegler. Relative Chebyshev centers in normed linear spaces. II. J. Approx. Theory, 38(4):293-311, 1983. · Zbl 0528.41020 [2] J. Bernal and F. Sullivan. Multidimensional volumes, super-reflexivity and normal structure in Banach spaces. Illinois J. Math., 27(3):501-513, 1983. 25 · Zbl 0512.46013 [3] M. S. Brodski˘ı and D. P. Mil 0 man. On the center of a convex set. Doklady Akad. Nauk SSSR (N.S.), 59:837-840, 1948. · Zbl 0030.39603 [4] J. A. Clarkson. Uniformly convex spaces. Trans. Amer. Math. Soc., 40(3):396-414, 1936. · Zbl 0015.35604 [5] M. M. Day. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Amer. Math. Soc., 47:313-317, 1941. · JFM 67.0402.04 [6] M. M. Day. Normed linear spaces. Springer-Verlag, New York-Heidelberg, third edition, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. · Zbl 0268.46013 [7] M. M. Day. Invariant renorming. In Fixed point theory and its applications (Proc. Sem., Dalhousie Univ., Halifax, N.S., 1975), pages 51-62. Academic Press, New York, 1976. · Zbl 0389.46007 [8] M. M. Day, R. C. James, and S. Swaminathan. Normed linear spaces that are uniformly convex in every direction. Canad. J. Math., 23, 1971. · Zbl 0229.46020 [9] H. Fakhoury. Directions d’uniforme convexit´ e dans un espace norm´ e. C. R. Acad. Sci. Paris S´ er. A-B, 283(7):Aiii, A473-A476, 1976. · Zbl 0332.46011 [10] T. Figiel. On the moduli of convexity and smoothness. Studia Math., 56(2):121-155, 1976. · Zbl 0344.46052 [11] A. L. Garkavi. On the optimal net and best cross-section of a set in a normed space. Izv. Akad. Nauk SSSR Ser. Mat., 26:87-106, 1962. · Zbl 0108.10801 [12] R. Geremia and F. Sullivan. Multidimensional volumes and moduli of convexity in Banach spaces. Ann. Mat. Pura Appl. (4), 127:231-251, 1981. · Zbl 0472.46018 [13] W. A. Kirk. A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly, 72:1004-1006, 1965. · Zbl 0141.32402 [14] W. A. Kirk. Nonexpansive mappings in product spaces, set-valued mappings and k-uniform rotundity. In Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983), volume 45 of Proc. Sympos. Pure Math., pages 51-64. Amer. Math. Soc., Providence, RI, 1986. · Zbl 0594.47048 [15] T. R. Landes. Normal structure and the sum-property. Pacific J. Math., 123(1):127-147, 1986. · Zbl 0629.46016 [16] T. C. Lim. On asymptotic centers and fixed points of nonexpansive mappings. Canad. J. Math., 32(2):421-430, 1980. · Zbl 0454.47045 [17] T. C. Lim, P.-K. Lin, C. Petalas, and T. Vidalis. Fixed points of isometries on weakly compact convex sets. J. Math. Anal. Appl., 282(1):1-7, 2003. · Zbl 1032.47036 [18] P. K. Lin. k-uniform rotundity is equivalent to k-uniform convexity. J. Math. Anal. Appl., 132(2):349-355, 1988. · Zbl 0649.46014 [19] V. D. Milman. Geometric theory of Banach spaces. II. Geometry of the unit ball. Uspehi Mat. Nauk, 26(6(162)):73-149, 1971. · Zbl 0229.46017 [20] R. R. Phelps. A representation theorem for bounded convex sets. Proc. Amer. Math. Soc., 11:976-983, 1960. 26 · Zbl 0098.07904 [21] S. Rajesh and P. Veeramani. Lim’s center and fixed point theorems for isometry mappings. Ann. Funct. Anal., To appear. · Zbl 1399.47132 [22] E. Silverman. Definitions of Lebesgue area for surfaces in metric spaces. Rivista Mat. Univ. Parma, 2:47-76, 1951. · Zbl 0043.05702 [23] I. Singer. Best approximation in normed linear spaces by elements of linear subspaces. Trans lated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wis senschaften, Band 171. Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. · Zbl 0197.38601 [24] M. A. Smith. Banach spaces that are uniformly rotund in weakly compact sets of directions. Canad. J. Math., 29(5):963-970, 1977. · Zbl 0338.46021 [25] M. A. Smith. Products of Banach spaces that are uniformly rotund in every direction. Pacific J. Math., 70(1):215-219, 1977. · Zbl 0373.46032 [26] M. A. Smith and B. Turett. A reflexive LUR Banach space that lacks normal structure. Canad. Math. Bull., 28(4):492-494, 1985. · Zbl 0546.46013 [27] F. Sullivan. A generalization of uniformly rotund Banach spaces. Canad. J. Math., 31(3):628- 636, 1979. · Zbl 0422.46011 [28] M. Veena Sangeetha and P. Veeramani. Normal structure and invariance of Chebyshev center under isometries. J. Math. Anal. Appl., 436(1):611-619, 2016. · Zbl 1338.46025 [29] V. Zizler. On some rotundity and smoothness properties of Banach spaces. Dissertationes Math. Rozprawy Mat., 87:33 pp., 1971. 27 · Zbl 0231.46036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.