Convex bodies #
This file contains the definition of the type
convex, compact, nonempty subsets of a real topological vector space
- define positive convex bodies, requiring the interior to be nonempty
- introduce support sets
- Characterise the interaction of the distance with algebraic operations, eg
dist (a • K) (a • L) = ‖a‖ * dist K L,
dist (a +ᵥ K) (a +ᵥ L) = dist K L
convex, convex body
V be a real topological vector space. A subset of
V is a convex body if and only if
it is convex, compact, and nonempty.
The convex bodies in a fixed space $V$ form a module over the nonnegative reals.
Convex bodies in a fixed seminormed space $V$ form a pseudo-metric space under the Hausdorff metric.
Convex bodies in a fixed normed space
V form a metric space under the Hausdorff metric.