N.I. Achieser, Vorlesungen über Approximationstheorie, Akademie-Verlag, Berlin, 1953.
Google Scholar
N.I. Achieser and M.G. Krein, Some Questions in the Theory of Moments, Transl. Math. Monogr., Vol. 2, American Mathematical Society, Providence, RI, 1962.
Google Scholar
A.R. Alimov, Solarity of sets in max-approximation problems, J. Fixed Point Theory Appl. 21 (2019), no. 3, Paper No. 76, 11 pp.
Google Scholar
A.R. Alimov and I.G. Tsarkov, Approximatively compact sets in asymmetric Efimov–Stechkin spaces and convexity of almost suns, Math. Notes 110 (2021), no. 5–6, 947–951.
Google Scholar
D. Amir, Chebyshev centers and uniform convexity, Pacific J. Math. 77 (1978), no. 1, 1–6.
Google Scholar
D. Amir and F. Deutsch, Suns, moons, and quasi-polyhedra, J. Approximation Theory 6 (1972), 176–201.
Google Scholar
E.Z. Andalafte and J.E. Valentine, Criteria for unique metric lines in Banach spaces, Proc. Amer. Math. Soc. 39 (1973), 367–370.
Google Scholar
K.W. Anderson, Midpoint Local Uniform Convexity, and Other Geometric Properties of Banach Spaces, PhD thesis, University of Illinois, Urbana, IL, 1960.
Google Scholar
E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel J. Math. 4 (1966), 213–216.
Google Scholar
A.A. Astaneh, A characterization of local uniform convexity of the norm, Indian J. Pure Appl. Math. 14 (1983), no. 10, 1217–1219.
Google Scholar
A.A. Astaneh, Completeness of normed linear spaces admitting centers, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 360–366.
Google Scholar
J.S. Bae and S.K. Choi, A note on k-uniformly convex spaces, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 3, 489–490.
Google Scholar
P. Bandyopadhyay, D. Huang, B.-L. Lin, and S.L. Troyanski, Some generalizations of locally uniform rotundity, J. Math. Anal. Appl. 252 (2000), no. 2, 906–916.
Google Scholar
P. Bandyopadhyay, Y. Li, B.-L. Lin, and D. Narayana, Proximinality in Banach spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 309–317.
Google Scholar
J. Blatter, Weiteste Punkte und n¨achste Punkte, Rev. Roumaine Math. Pures Appl. 14 (1969), 615–621.
Google Scholar
J. Blatter, P.D. Morris, and D.E. Wulbert, Continuity of the set-valued metric projection, Math. Ann. 178 (1968), 12–24.
Google Scholar
B. Brosowski and F. Deutsch, Radial continuity of set-valued metric projections, J. Approximation Theory 11 (1974), 236–253.
Google Scholar
H. Busemann, Note on a theorem of convex sets, Mat. Tidsskr. B 1947 (1947), 32–34.
Google Scholar
U.S. Chakraborty, On a generalization of local uniform rotundity, arXiv preprint, 2020. Available at arXiv: 2001.00696.
Google Scholar
Q.J. Cheng, B. Wang, and C.L. Wang, On uniform convexity of Banach spaces, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 3, 587–594.
Google Scholar
J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414.
Google Scholar
S. Cobzaş, Geometric properties of Banach spaces and the existence of nearest and farthest points, Abstr. Appl. Anal. 2005, no. 3, 259–285.
Google Scholar
D.F. Cudia, Rotundity, in: V.L. Klee (ed.), Proceedings of Symposia in Pure Mathematics. Vol. VII: Convexity, American Mathematical Society, Providence, RI, 1963, pp. 73–97.
Google Scholar
G.R. Damai and P.M. Bajracharya, Uniformly rotund in every direction (URED) norm, Int. J. Sci. Res. Pub. 6 (2016), no. 9, 74–83.
Google Scholar
M.M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313–317.
Google Scholar
M.M. Day, Some more uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 504–507.
Google Scholar
M.M. Day, Uniform convexity. III, Bull. Amer. Math. Soc. 49 (1943), 745–750.
Google Scholar
M.M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math. (2) 45 (1944), 375–385.
Google Scholar
M.M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.
Google Scholar
M.M. Day, Normed Linear Spaces, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958.
Google Scholar
M.M. Day, R.C. James, and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canadian J. Math. 23 (1971), 1051–1059.
Google Scholar
D. Delbosco and F. Rossati, Characterizations of strictly convex normed linear spaces, Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2 (1981/82), no. 1, 359–369.
Google Scholar
F. Deutsch and J.M. Lambert, On continuity of metric projections, J. Approx. Theory 29 (1980), no. 2, 116–131.
Google Scholar
R. Deville and V.E. Zizler, Farthest points in w*-compact sets, Bull. Austral. Math. Soc. 38 (1988), no. 3, 433–439.
Google Scholar
S. Dutta and P Shunmugaraj, Strong proximinality of closed convex sets, J. Approx. Theory 163 (2011), no. 4, 547–553.
Google Scholar
S. Dutta and P. Shunmugaraj, Weakly compactly LUR Banach spaces, J. Math. Anal. Appl. 458 (2018), no. 2, 1203–1213.
Google Scholar
M. Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176.
Google Scholar
M. Edelstein, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Soc. 43 (1968), 375–377.
Google Scholar
M. Edelstein, Weakly proximinal sets, J. Approximation Theory 18 (1976), no. 1, 1–8.
Google Scholar
N.V. Efimov and S.B. Stechkin, Approximative compactness and Chebyshev sets, Dokl. Akad. Nauk SSSR 140 (1961), 522–524.
Google Scholar
K. Eshita and W. Takahashi, On the uniform convexity of subsets of Banach spaces, Sci. Math. Jpn. 60 (2004), no. 4, 577–594.
Google Scholar
K. Fan and I. Glicksberg, Fully convex normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 947–953.
Google Scholar
K. Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25 (1958), 553–568.
Google Scholar
S. Fitzpatrick, Metric projections and the differentiability of distance functions, Bull. Austral. Math. Soc. 22 (1980), no. 2, 291–312.
Google Scholar
J. Fletcher and W.B. Moors, Chebyshev sets, J. Aust. Math. Soc. 98 (2015), no. 2, 161–231.
Google Scholar
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 (1962), 87–106.
Google Scholar
P. Gayathri and V. Thota, Characterizations of weakly uniformly rotund Banach spaces, J. Math. Anal. Appl. 514 (2022), no. 1, Paper No. 126298, 15 pp.
Google Scholar
S. Gudder and D. Strawther, Strictly convex normed linear spaces, Proc. Amer. Math. Soc. 59 (1976), no. 2, 263–267.
Google Scholar
A.J. Guirao and V. Montesinos, A note in approximative compactness and continuity of metric projections in Banach spaces, J. Convex Anal. 18 (2011), no. 2, 397–401.
Google Scholar
S. Gupta and T.D. Narang, Strong proximinality and rotundities in Banach spaces, J. Adv. Math. Stud. 10 (2017), no. 2, 174–182.
Google Scholar
P. Hájek and A. Quilis, Counterexamples in rotundity of norms in Banach spaces, arXiv preprint, 2023. Available at arXiv: 2302.11041.
Google Scholar
R.A. Hirschfeld, On best approximations in normed vector spaces. II, Nieuw Arch. Wisk. (3) 6 (1958), 99–107.
Google Scholar
R.B. Holmes, A Course on Optimization and Best Approximation, Lecture Notes in Math., Vol. 257, Springer-Verlag, Berlin-New York, 1972.
Google Scholar
R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749.
Google Scholar
V.I. Istrățescu, Fixed Point Theory, Math. Appl., 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981.
Google Scholar
V.I. Istrățescu, Strict Convexity and Complex Strict Convexity, Lecture Notes in Pure and Appl. Math., 89, Marcel Dekker, Inc., New York, 1984.
Google Scholar
V.I. Istrățescu and J.R. Partington, On nearly uniformly convex and k-uniformly convex spaces, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 325–327.
Google Scholar
P.S. Kenderov, Uniqueness on a residual part of best approximations in Banach spaces, Pliska Stud. Math. Bulgar. 1 (1977), 122–127.
Google Scholar
V.L. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43.
Google Scholar
V.L. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), pp. 51–63.
Google Scholar
V.L. Klee, Convexity of Chebyshev sets, Math. Ann. 142 (1961), 292–304.
Google Scholar
S.V. Konyagin, Sets of points of nonemptiness and continuity of a metric projection, Mat. Zametki 33 (1983), no. 5, 641–655.
Google Scholar
K.-S. Lau, Farthest points in weakly compact sets, Israel J. Math. 22 (1975), no. 2, 168–174.
Google Scholar
K.-S. Lau, Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J. 27 (1978), no. 5, 791–795.
Google Scholar
K.-S. Lau, Best approximation by closed sets in Banach spaces, J. Approx. Theory 23 (1978), no. 1, 29–36.
Google Scholar
B.L. Lin and X.T. Yu, On the k-uniform rotund and the fully convex Banach spaces, J. Math. Anal. Appl. 110 (1985), no. 2, 407–410.
Google Scholar
A.R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225–238.
Google Scholar
R.E. Megginson, The Semi-Kadec-Klee Condition and Nearest-Point Properties of Sets in Normed Linear Spaces, PhD thesis, University of Illinois, Urbana, IL, 1984.
Google Scholar
R.E. Megginson, An Introduction to Banach Space Theory, Grad. Texts in Math., 183, Springer-Verlag, New York, 1998.
Google Scholar
H.N. Mhaskar and D.V. Pai, Fundamentals of Approximation Theory, Narosa Publishing House, New Delhi, 2000.
Google Scholar
V.D. Milman, Geometric theory of Banach spaces. Part II. Geometry of the unit sphere, Russian Math. Surveys 26 (1971), no. 6, 79–163.
Google Scholar
S. Miyajima and F. Wada, Uniqueness of a farthest point in a bounded closed set in Banach spaces, SUT J. Math. 29 (1993), no. 2, 291–310.
Google Scholar
A. Moltó, J. Orihuela, S. Troyanski, and M. Valdivia, On weakly locally uniformly rotund Banach spaces, J. Funct. Anal. 163 (1999), no. 2, 252–271.
Google Scholar
T.D. Narang and S. Gupta, On Chebyshev centers, Bull. Allahabad Math. Soc. 34 (2019), no. 2, 181–199.
Google Scholar
E.V. Oshman, A continuity criterion for metric projections in Banach spaces, Math. Notes Acad. Sci. USSR 10 (1971), no. 4, 697–701.
Google Scholar
B.B. Panda and O.P. Kapoor, Approximative compactness and continuity of metric projections, Bull. Austral. Math. Soc. 11 (1974), 47–55.
Google Scholar
B.B. Panda and O.P. Kapoor, A generalization of local uniform convexity of the norm, J. Math. Anal. Appl. 52 (1975), no. 2, 300–308.
Google Scholar
B.B. Panda and O.P. Kapoor, On farthest points of sets, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 10, 1369–1377.
Google Scholar
B.B. Panda and O.P. Kapoor, On farthest points of sets, J. Math. Anal. Appl. 62 (1978), no. 2, 345–353.
Google Scholar
R.R. Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc. 8 (1957), 790–797.
Google Scholar
T. Polak and B. Sims, A Banach space which is fully 2-rotund but not locally uniformly rotund, Canad. Math. Bull. 26 (1983), no. 1, 118–120.
Google Scholar
V.S. Raj and A.A. Eldred, A characterization of strictly convex spaces and applications, J. Optim. Theory Appl. 160 (2014), no. 2, 703–710.
Google Scholar
J.P. Revalski and N.V. Zhivkov, Best approximation problems in compactly uniformly rotund spaces, J. Convex Anal. 19 (2012), no. 4, 1153–1166.
Google Scholar
D. Sain, V. Kadets, K. Paul, and A. Ray, Chebyshev centers that are not farthest points, Indian J. Pure Appl. Math. 49 (2018), no. 2, 189–204.
Google Scholar
D. Sain, K. Paul, and A. Ray, Farthest point problem and M-compact sets, J. Nonlinear Convex Anal. 18 (2017), no. 3, 451–457.
Google Scholar
I. Singer, Some remarks on approximative compactness, Rev. Roumaine Math. Pures Appl. 9 (1964), 167–177.
Google Scholar
I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York-Berlin, 1970.
Google Scholar
M.A. Smith, Banach spaces that are uniformly rotund in weakly compact sets of directions, Canadian J. Math. 29 (1977), no. 5, 963–970.
Google Scholar
M.A. Smith, Products of Banach spaces that are uniformly rotund in every direction, Pacific J. Math. 70 (1977), no. 1, 215–219.
Google Scholar
M.A. Smith, A reflexive Banach space that is LUR and not 2R, Canad. Math. Bull. 21 (1978), no. 2, 251–252.
Google Scholar
M.A. Smith, Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), no. 2, 155–161.
Google Scholar
M.A. Smith, A Banach space that is MLUR but not HR, Math. Ann. 256 (1981), no. 2, 277–279.
Google Scholar
V.L. Šmulian, On some geometrical properties of the unit sphere in the space of the type (B), Rec. Math. N.S. [Mat. Sbornik] 6/48 (1939), 77–94.
Google Scholar
V.L. Šmulian, Sur la dérivabilité de la norme dans l’espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N.S.) 27 (1940), 643–648.
Google Scholar
V.L. Šmulian, Sur la structure de la sphère unitaire dans l’espace de Banach, Rec. Math. [Mat. Sbornik] N.S. 9/51 (1941), 545–561.
Google Scholar
S.B. Stechkin, Approximation properties of sets in linear normed spaces, Rev. Math. Pures Appl. 8 (1963), 5–18.
Google Scholar
F. Sullivan, A generalization of uniformly rotund Banach spaces, Canadian J. Math. 31 (1979), no. 3, 628–636.
Google Scholar
I.G. Tsarkov, Uniform convexity in nonsymmetric spaces, Math. Notes 110 (2021), no. 5–6, 773–783.
Google Scholar
L.P. Vlasov, Chebyshev sets and approximately convex sets, Math. Notes Acad. Sci. USSR 2 (1967), no. 2, 600–605.
Google Scholar
L.P. Vlasov, On Chebyshev sets, Soviet Math. Dokl. 8 (1967), 401–404.
Google Scholar
L.P. Vlasov, Chebyshev sets and some generalizations of them, Math. Notes Acad. Sci. USSR 3 (1968), no. 1, 36–41.
Google Scholar
L.P. Vlasov, Approximative properties of sets in normed linear spaces, Russian Math. Surveys 28 (1973), no. 6, 1–66.
Google Scholar
R. Vyborny, On the weak convergence in locally uniformly convex spaces, Casopis Pěst. Mat. 81 (1956), no. 3, 352–353.
Google Scholar
A.O.Wanjara, A classical survey on rotundity of norms in Banach spaces, Indonesian J. Math. Appl. 2 (2024), no. 1, 42–57.
Google Scholar
C.X. Wu and Y.J. Li, Strong convexity in Banach spaces, J. Math. (Wuhan) 13 (1993), no. 1, 105–108.
Google Scholar
D.E. Wulbert, Continuity of metric projections, Trans. Amer. Math. Soc. 134 (1968), 335–341.
Google Scholar
M. Zhao, B.X. Fang, and Y.H.Wang, K-rotundities and their generalizations, Chinese Ann. Math. Ser. A 21 (2000), no. 3, 289–294.
Google Scholar
N.V. Zhivkov, Metric projections and antiprojections in strictly convex normed spaces, C. R. Acad. Bulgare Sci. 31 (1978), no. 4, 369–372.
Google Scholar
N.V. Zhivkov, Continuity and nonmultivaluedness properties of metric projections and antiprojections, Serdica 8 (1982), no. 4, 378–385.
Google Scholar
V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 33 pp. (errata insert).
Google Scholar
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