The Minkowski functional #
This file defines the Minkowski functional, aka gauge.
The Minkowski functional of a set
s is the function which associates each point to how much you
need to scale
x to be inside it. When
s is symmetric, convex and absorbent, its gauge is
a seminorm. Reciprocally, any seminorm arises as the gauge of some set, namely its unit ball. This
induces the equivalence of seminorms and locally convex topological vector spaces.
Main declarations #
For a real vector space,
gauge: Aka Minkowski functional.
gauge s xis the least (actually, an infimum)
x ∈ r • s.
gaugeSeminorm: The Minkowski functional as a seminorm, when
sis symmetric, convex and absorbent.
- [H. H. Schaefer, Topological Vector Spaces][schaefer1966]
Minkowski functional, gauge
s is a convex neighborhood of the origin in a topological real vector space, then
is continuous. If the ambient space is a normed space, then
gauge s is Lipschitz continuous, see