  Given any finite set of nonnegative integers, there exists a closed convex
set whose facial dimension signature coincides with this set of integers, that
is, the dimensions of its nonempty faces comprise exactly this set of integers.
In this work, we show that such sets can be realised as solution sets of
systems of finitely many convex quadratic inequalities, and hence are
representable via second-order cone programming problems, and are, in
particular, spectrahedral. It also follows that these sets are facially
exposed, in contrast to earlier constructions. We obtain a lower bound on the
minimum number of convex quadratic inequalities needed to represent a closed
convex set with prescribed facial dimension signature, and show that our bound
is tight for some special cases. Finally, we relate the question of finding
efficient representations with indecomposability of integer sequences and other
topics, and discuss a substantial number of open questions.
