Regular points of extremal subsets in Alexandrov spaces

We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighbor-hood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal sub-set has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regular part of an extremal subset. In an appendix, a deformation retraction of a metric neighborhood of a general extremal subset is constructed.