  We define regular points of an extremal subset in an Alexandrov space and
study their basic properties. We show that a neighborhood 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 subset 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.
