  This paper is concerned with signal detection in Gaussian noise under
quadratically convex orthosymmetric (QCO) constraints. Specifically the null
hypothesis assumes no signal, whereas the alternative considers signal which is
separated in Euclidean norm from zero, and belongs to the QCO constraint. Our
main result establishes the minimax rate of the separation radius between the
null and alternative purely in terms of the geometry of the QCO constraint --
we argue that the Kolmogorov widths of the constraint determine the critical
radius. This is similar to the estimation problem with QCO constraints, which
was first established by Donoho et al. (1990); however, as expected, the
critical separation radius is smaller compared to the minimax optimal
estimation rate. Thus signals may be detectable even when they cannot be
reliably estimated.
