quickfur wrote:I'm thinking in terms of "native" hyperbolic polytopes. I.e., polytopes that exist in hyperbolic space, using the native (i.e., hyperbolic) definition of straight lines to determine convexity.

For example, in Euclidean 2D plane, a vertex surrounded by 4 squares has zero angle defect, so it cannot be strictly convex. In hyperbolic space, however, a vertex surrounded by 4 quadrilaterals will have positive angle defect, and so can be part of a larger strictly-convex polytope.

In the Klein model (corresponding to gnomonic projection), hyperbolic straight lines are the same as Euclidean straight lines, so convexity is the same. But the angles are different. In particular, a hyperbolic convex regular polygon will generally be projected to a convex irregular polygon.

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