  We define and study a class of subshifts of finite type (SFTs) defined by a
family of allowed patterns of the same shape where, for any contents of the
shape minus a corner, the number of ways to fill in the corner is the same. The
main results are that for such an SFT, a locally legal pattern of convex shape
is globally legal, and there is a measure that samples uniformly on all convex
sets. Under suitable computability assumptions, this measure can be sampled,
and legal configurations counted and enumerated, effectively and efficiently.
We show by example that these subshifts need not admit a group (more generally
unital magma or quasigroup) structure by shift-commuting continuous operations.
Our approach to convexity is axiomatic, and only requires an abstract convex
geometry that is "midpointed with respect to the shape". We construct such
convex geometries on several groups, in particular all strongly polycyclic
groups and free groups. We also show some other methods for sampling finite
patterns, one based on orderings and one based on contructing new "independent
sets" from old. We also show a link to conjectures of Gottshalk and Kaplansky.
