Introduce common interfaces (abstract supertypes) for similar types about lazysets.jl HOT 3 CLOSED

Proposal (I am open for different names):

ZeroSet, Singleton <: SingleSet  # provide: element(dim), element
HPolygon, HPolygonOpt <: HPolygonal  # common: many methods
HPolygonal, VPolygon <: Polygonal  # provide: tovrep, tohrep; inherit: dim
SingleSet, BallInf, Hyperrectangle <: Box  # provide: radius_b(dim), radius_b
Ball, Box, Zonotope <: PointSymmetric <: LazySet  # provide: center; inherit: dim
Ball1, Box, Polygonal, Zonotope <: Polytopic <: LazySet  # provide: vertices_list

[SingleSet, Ball1, Ball2, Ballp, BallInf <: Ball] -- removed for simplicity

radius_b should not be confused with the existing radius for general LazySets.

from lazysets.jl.

Another possible extension is to have implementations for "centered in the origin" sets such as a Hyperrectangle0. These are more efficient to store and process (no center).
If we want this, we should introduce another layer of abstract types for each concrete type. Then we can provide different implementations for the same concepts. We already have this for HPolygon and HPolygonOpt.

We do not have to implement such sets right away, but with these changes this will be straight-forward to integrate them later.

from lazysets.jl.

Noooo, Julia does not support multiple supertypes (yet).

One can simulate multiple (abstract) supertypes with artificial combination types of course. Maybe I will just do that... 😈

from lazysets.jl.