  We give a constructive proof for the following new collar theorem: every
locally collared closed set that is paracompact in a Hausdorff space is
collared. This includes the important special case of locally collared closed
sets in paracompact Hausdorff spaces. Importantly, we use Stone's result that
every open cover of a paracompact space has an open locally finite refinement
which is the countable union of discrete families. Furthermore, in the LIP
category, our construction yields collars that are locally bi-Lipschitz
embeddings. If the initial data satisfy uniform estimates, then this collar is
even bi-Lipschitz onto its image and we explicitly bound the constants. We also
provide partitions of unity whose Lipschitz constants are bounded by the
Lebesgue constant and the order of the cover.
