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.

Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to $(X,D\left({(-A)}^{\alpha }\right))$, α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, ${\left|\right|\lambda R(\lambda ;A)f\left|\right|}_Y{\le c\phi (1/\lambda )\left|\right|f\left|\right|}_X$, for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has good control...

In [6], C. Dierick deals with a small but important collection of norms in the product of a finite number of normed linear spaces and he extends to such products some results on functional characterization of best approximations. In this paper we establish the widest scope in which the mentioned results remain valid.