  We study incompatibility of measurements and its relation to steering and
nonlocality in a class of finite dimensional general probabilistic theories
(GPT).The basic idea is to represent finite collections of measurements as
affine maps of a state space into a polysimplex and show that incompatibility
is characterized by properties of these maps. We introduce the notion of an
incompatibility witness and show its relation to incompatibility degree. We
find the maximum incompatibility degree attainable by pairs of two-outcome
quantum measurements and characterize state spaces for which incompatibility
degree attains maximal values possible in GPT. As examples, we study the spaces
of classical and quantum channels and show their close relation to
polysimplices. This relation explains the super-quantum non-classical effects
that were observed on these spaces.
