Semialgebraic Whitney extension problem and metric geometry of singularities

Semialgebraic Whitney extension problem asks the following. Given a semialgebraic function
defined on a semialgebraic subset of Rn that admits a Cm extension to Rn. Does it admit a semialgebraic Cm extension? The answer (always affirmative so far) is known only in some special cases. A simpler analogous problem of semialgebraic extension of Whitney fields has been answered affirmatively by Kurdyka and Pawłucki.
In this talk I present metric properties of semialgebraic sets and, in more generality, sets definable in o-minimal structures, that are used to obtained some of these partial answers. In particular the one of Kurdyka and Pawłucki and a recent solution by Fefferman and Luli of the original problem, and the related semialgebraic linear equation problem, for n = 2.