"Convex" polytopes in hyperbolic space?

[quickfur]

I recently started dabbling with hyperbolic geometry. Started grokking stuff like equidistant curves, horocycles, ultraparallel lines, etc..

This has made me start wondering, is it possible to define convexity in hyperbolic space in a consistent way? If so, does it increase the set of convex {regular, uniform, etc.} polytopes in n dimensions?

[Plasmath]

The formal definition for convex is that given any two points in the shape they can connect with a line segment that does not leave the shape. Because we can define lines in hyperbolic space, we can therefore define line segments, so we can therefore we can determine convexity.
Two other related things: H³ and H²R¹ spaces could have different convex polytopes,
And more importantly, every two points has multiple lines fitting through it, so there are two possible definitions that I think could work:
At least one line segment stays within the shape
All possible line segments stay within the shape
Not sure what new convex polytopes exist, however.

[quickfur]

But I thought in Hⁿ, there's a unique shortest line segment between every pair of points? (Even if there are multiple lines that could fit through it.)

[Klitzing]

The facets of any hyperbolic Wythoffian with integral CD link marks only will be convex.
--- rk

[quickfur]

[...]
And more importantly, every two points has multiple lines fitting through it
[...]

Hold on a sec, isn't that wrong? I thought only in spherical space there are multiple lines through a fixed pair of points. In hyperbolic space there should only be a single line through any pair of points?

[Marek14]

[...]
And more importantly, every two points has multiple lines fitting through it
[...]

Hold on a sec, isn't that wrong? I thought only in spherical space there are multiple lines through a fixed pair of points. In hyperbolic space there should only be a single line through any pair of points?

Yes. And even in spherical space, multiple lines exist only for antipodal points -- changing to elliptic space fixes that problem.

Multiple geodesics connecting two points is more a problem in quotient or Nil.

[quickfur]

Does that mean that hyperbolic space does not introduce any "new" convex polytopes that doesn't already exist in Euclidean space?

[Marek14]

Does that mean that hyperbolic space does not introduce any "new" convex polytopes that doesn't already exist in Euclidean space?

Yup, though there are some more "strictly convex". For example, a line of cubes in Euclidean space has 180-degree angles between them, but in hyperbolic space, it's less.

[quickfur]

Does that mean that hyperbolic space does not introduce any "new" convex polytopes that doesn't already exist in Euclidean space?

Yup, though there are some more "strictly convex". For example, a line of cubes in Euclidean space has 180-degree angles between them, but in hyperbolic space, it's less.

But doesn't that mean there ought to be more convex polytopes in hyperbolic space? Because you could have a fragment of the surface consist of, say, this line of cubes, and it would be convex where in Euclidean space it would be degenerate. So there ought to be more strictly convex polytopes?

[Marek14]

Does that mean that hyperbolic space does not introduce any "new" convex polytopes that doesn't already exist in Euclidean space?

Yup, though there are some more "strictly convex". For example, a line of cubes in Euclidean space has 180-degree angles between them, but in hyperbolic space, it's less.

But doesn't that mean there ought to be more convex polytopes in hyperbolic space? Because you could have a fragment of the surface consist of, say, this line of cubes, and it would be convex where in Euclidean space it would be degenerate. So there ought to be more strictly convex polytopes?

Well, this all depends on how do you count polytopes. When do you consider an Euclidean and hyperbolic polytope analogical, exactly?

After all, you can put a hyperbolic polytope inside a sphere in Klein projection, and the projection will then be an Euclidean polytope.

[quickfur]

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.

[Plasmath]

This may not be what you're looking for, but ideal polyhedra/polytopes should be strictly convex. Adjusting the vertices to not be ideal points should give finite strictly convex hyperbolic polytopes, but I haven't checked yet.

... does it [hyperbolic space] increase the set of convex {regular, uniform, etc.} polytopes in n dimensions?

Ideal polyhedra don't really add any new polytopes. These really just create hyperbolic versions of existing polytopes and don't create new ones.

[wendy]

No additional uniform convex polytopes exist in hyperbolic space, that isn't already in euclidean space.

On the other hand John Conway and I worked on various different and inter-related methods to describe the infinitude of those that exist as convex polytopes in H3.

[mr_e_man]

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.

[wendy]

You can, for example, join any number of dodecahedra into a convex polytope, as long as they come from {5,3,5} and there are not three at an edge. Such figures are yicloid fragments, roughly correspond to thinking that polynimos can be made into convex polyhedra. The usual restriction to stop this is to suppose that a common circumradius must also exist.