  Recent advances in statistics introduced versions of the central limit
theorem for high-dimensional vectors, allowing for the construction of
confidence regions for high-dimensional parameters. In this note, $s$-sparsely
convex high-dimensional confidence regions are compared with respect to their
volume. Specific confidence regions which are based on $\ell_p$-balls are found
to have exponentially smaller volume than the corresponding hypercube. The
theoretical results are validated by a comprehensive simulation study.
