  We propose a measure of quantum steerability, namely a convex steering
monotone, based on the trace distance between a given assemblage and its
corresponding closest assemblage admitting a local-hidden-state (LHS) model. We
provide methods to estimate such a quantity, via lower and upper bounds, based
on semidefinite programming. One of these upper bounds has a clear geometrical
interpretation as a linear function of rescaled Euclidean distances in the
Bloch sphere between the normalized quantum states of: (i) a given assemblage
and (ii) an LHS assemblage. For a qubit-qubit quantum state, the above ideas
also allow us to visualize various steerability properties of the state in the
Bloch sphere via the so-called LHS surface. In particular, some steerability
properties can be obtained by comparing such an LHS surface with a
corresponding quantum steering ellipsoid. Thus, we propose a witness of
steerability corresponding to the difference of the volumes enclosed by these
two surfaces. This witness (which reveals the steerability of a quantum state)
enables finding an optimal measurement basis, which can then be used to
determine the proposed steering monotone (which describes the steerability of
an assemblage) optimized over all mutually-unbiased bases.
