Let $X$ be a non-reflexive real Banach space. Then for each norm $|·|$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|·\left|\right)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

We study the notion of fractional $L^p$-differentiability of order $s\in (0,1)$ along vector fields satisfying the Hörmander condition on ${ℝ}^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p}\subset W^{s,q}$, where q is a suitable exponent greater than p.

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher smoothness (C∞) is involved. In this note we survey most of the main results in this area, and indicate many old as well as new open problems.