  Given a spline space spanned by Truncated Hierarchical B-splines (THB), it is
always possible to construct a spline space spanned by Locally Refined
B-splines (LRB) that contains the THB-space. Starting from configurations where
the two spline spaces are equal, we adress what happens to the properties of
the LRB-space when it is modified by local one-directional refinement at convex
corners of, and along edges between dyadic refinement regions. We show that
such local modifications can reduce the number of B-splines over each element
to the minimum prescribed by the polynomial bi-degree, and that such local
refinements can be used for improving the condition numbers of mass and
stiffness matrices.
