  The sign uncertainty principle of Bourgain, Clozel & Kahane asserts that if a
function $f:\mathbb{R}^d\to \mathbb{R}$ and its Fourier transform $\widehat{f}$
are nonpositive at the origin and not identically zero, then they cannot both
be nonnegative outside an arbitrarily small neighborhood of the origin. In this
article, we establish some equivalent formulations of the sign uncertainty
principle, and in particular prove that minimizing sequences exist within the
Schwartz class when $d=1$. We further address a complementary sign uncertainty
principle, and show that corresponding near-minimizers concentrate a universal
proportion of their positive mass near the origin in all dimensions.
