  Numerical radius $r(A)$ is the radius of the smallest ball with the center at
zero containing the field of values of a given square matrix $A$. It is well
known that $r(A)\leq \|A\| \leq 2r(A)$, where $\| \cdot \|$ is the matrix
2-norm. Matrices attaining the lower bound are called radial, and have been
analyzed thoroughly. This is not the case for matrices attaining the upper
bound where only partial results are available. In this paper we consider
matrices satisfying $r(A)=\|A\|/2$, and call them half-radial. We summarize the
existing results and formulate new ones. In particular, we investigate their
singular value decomposition and algebraic structure, and provide other
necessary and sufficient conditions for a matrix to be half-radial. Based on
that, we study the extreme case of the attainable constant~$2$ in Crouzeix's
conjecture. The presented results support the conjecture of Greenbaum and
Overton, that the Crabb-Choi-Crouzeix matrix always plays an important role in
this extreme case.
