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We provide an elementary proof of a result by V. P. Fonf and C. Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J. Lindenstrauss and R. R. Phelps [Israel J. Math. 6 (1968), pp. 39–48] to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result. Let X be an infinite-dimensional Hilbert space satisfying dens(X)< 2ℵ0, then X does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V. P. Fonf, M. Levin, and C. Zanco in [J. Geom. Anal. 24 (2014), pp. 1891–1897] to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth. [ABSTRACT FROM AUTHOR]

Geremia and Sullivan [Ann. Math Pure Appl.127 (1981), 231–251] gave a necessary and sufficient condition for an ℓp-product of spaces to be 2-uniformly rotund. We extend this result to k-uniform rotundity for any integer k > 1. The nonreflexive, uniformly nonoctahedral Banach space X ˜ constructed by James [Israel J. Math.18 (1974), 145–155] does not contain arbitrarily close copies of although it is not k-uniformly rotund (k-UR) for any k ⩾ 2. This shows that a Banach space X not being k-UR does not imply that X contains arbitrarily close copies of ℓ1k+1 for each k ⩾ 2. We show that a sufficient condition that a Banach space X be not k-UR is that it contains an arbitrarily close copy of one of the faces of ℓ1k+1 rather than itself. [ABSTRACT FROM AUTHOR]