On uniform canonical bases in $L_p$ lattices and other metric structures

We discuss the notion of uniform canonical bases, both in an abstract manner and specifically for the theory of atomless Lp lattices. We also discuss the connection between the definability of the set of uniform canonical bases and the existence of the theory of beautiful pairs (i.e., with the finite cover property), and prove in particular that the set of uniform canonical bases is definable in algebra- ically closed metric valued fields. 2010 Mathematics Subject Classification 03C45 (primary); 46B42,12J25 (second- ary)