[rgvdheir] random convex polytope

first, create a random smooth shape by picking random weights for the first few spherical harmonics and summing them.  reject if the object is not convex: determining this seems difficult.  can the coefficients be chosen to favor convexity?  previously 2D.

then, uniformly randomly sample some points on the surface and construct tangent planes, forming a random convex polyhedron (might not be closed).  uniform sampling of surface points on an arbitrary surface seems difficult.  maybe a random walk of small steps each of approximately constant geodesic length.  how many steps?

or, permit starting with a smooth shape not necessarily convex.  reject a tangent plane if its inward half space does not contain the origin.

extend this construction to higher dimensions.  what are the hyper-spherical harmonics?  curse of dimensionality probably somehow happens.

a 4D polytope (polychoron) is interesting because its cells are random polyhedra whose faces fit together pairwise but not further in 3D.  how can a collection of specified irregular polyhedra be manufactured?  3D printing is of course one way.  or, mill some faces, but include nubs on one or more faces.  then, reorient the piece upside down, holding it in place from below by the nubs.  mill the remaining faces.  then, somehow remove the nubs.

mill two negative halves then cast then join somehow?

create a net and fold (origami)?

the edges of a random convex 3D polyhedron define a random planar graph.  do analogous graphs for convex polytopes in higher dimensions have a property analogous to planarity?

Delaunay triangulation is another way to get a random planar graph, but faces are generally always triangles.  but are most of the faces produced by the tangent method described above also triangles?

3D convex hull of random points results in a convex polyhedron, and this generalizes to arbitrary dimensions.  I think most faces will again be triangles (simplices).  can the points be sampled in a manner to get interesting shapes?  for example, after the first few points, avoid new points that are way outside the convex hull so far, drastically changing the polyhedron.

aside: the minimal surface in 3D that contains all given points is not necessarily the convex hull as it is in 2D.  the minimal surface might have hyperbolic paraboloid faces.

a random convex polytope generates a random linear or quadratic programming problem.

applying a linear transformation to a convex polytope maintains its convexity.  proof: assume false, meaning that after the linear transformation there is a line segment that starts inside, goes outside, then comes back inside (definition of non-convex).  apply the inverse linear transformation, which is linear.  the line segment remains a line segment.  it still goes in out in.  the existence of such a line segment in the initial polytope means it was not convex.

consider relaxing the requirement of convexity and only require non-self-intersecting.  one can have holes and other weird features, though perhaps some so weird that it might be good to forbid them.  what weird features, especially in higher dimensions?  polygonal faces can be non-convex with holes.  that is probably OK, but faces with intersecting edges (e.g., pentagram) are not OK.

previous thoughts on random polyhedra..