  In this work, we characterize the amount of steerability present in quantum
theory by connecting the maximal violation of a steering inequality to an
inclusion problem of free spectrahedra. In particular, we show that the maximal
violation of an arbitrary unbiased dichotomic steering inequality is given by
the inclusion constants of the matrix cube, which is a well-studied object in
convex optimization theory. This allows us to find new upper bounds on the
maximal violation of steering inequalities and to show that previously obtained
violations are optimal. In order to do this, we prove lower bounds on the
inclusion constants of the complex matrix cube, which might be of independent
interest. Finally, we show that the inclusion constants of the matrix cube and
the matrix diamond are the same. This allows us to derive new bounds on the
amount of incompatibility available in dichotomic quantum measurements in fixed
dimension.
