Are there convexity generalizations that admit genus information?
For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can be joined by a line or a circle seems to work.
Is there a good definition that works in higher dimensions for which appropriate generalization of traditional convex geometry inequalities such as Brunn-Minkowski inequality can be given (A suitable notion of convex hull needs to be first defined).
Natural higher genus analogues of convex surfaces are usually considered to be surfaces which satisfy the "two piece property" or are "tight".
A closed surface in Euclidean space is said to have the two piece property or be tight if no plane cuts it into more than two pieces. So for instance tori of revolution are tight as are all convex surfaces, and the polyhedral torus referenced in the question. An equivalent formulation is that the "total positive curvature", or the integral of the Gauss curvature where it is positive, must be 4π, which is its lowest possible value. This implies that all points of positive curvature must lie on the boundary of the convex hull of the surface.
The study of these surfaces originates with Alexandrov, who showed that tight analytic surfaces are isometrically rigid. Later significant contributions were made by Chern and Lashoff (who studied total curvature), Nirenberg (Who tried to extend Alexandrov's rigidity theorem to smooth surfaces), Banchoff (who coined the term "two piece property"), Kuiper, and others.
A very good reference is the survey book Tight and Taut Subamnifolds, edited by Cecil and Chern.
Just a note to point out that not every pair of points can be joined by a line or a circular arc if the hexagonal torus ("this polyhedron") is thin (inner and outer radii are close to one another). For then the circular arcs cannot always remain inside the torus.
