In this paper we provide an answer to a question found in "Groupoid metrization theory," With applications to analysis on quasi-metric spaces and functional analysis,, namely when given a quasi-metric $\rho$, if one examines all quasi-metrics which are pointwise equivalent to $\rho$, does there exist one which is most like an ultrametric (or, equivalently, exhibits an optimal amount of Hölder regularity)? The answer, in general, is negative, which we demonstrate by constructing a suitable Rolewicz--Orlicz space.
"Optimal Quasi-Metrics in a Given Pointwise Equivalence Class do not Always Exist." Publ. Mat. 59 (2) 479 - 509, 2015.