[Paper Review] Some remarks on convex analysis in topological groups

Jonathan M. Borwein, Ohad Giladi|arXiv (Cornell University)|Oct 15, 2015

Advanced Topology and Set Theory11 references3 citations

TL;DR

This paper extends classical convex analysis to topological groups and monoids by introducing a group-theoretic gauge functional and proving separation, Krein-Milman, and minimax theorems in locally convex topological groups. It establishes that convexity and duality results from vector spaces generalize under mild topological and algebraic conditions, particularly in p-semidivisible or divisible monoids with precompact convex hulls.

ABSTRACT

We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.

Motivation & Objective

• To extend fundamental results of convex analysis—such as separation theorems and the Krein-Milman theorem—to the setting of topological groups and monoids.
• To define and analyze a group-theoretic analogue of the gauge functional used in locally convex topological vector spaces.
• To establish conditions under which minimax theorems hold in topological monoids, particularly in p-semidivisible or divisible structures.
• To investigate the role of convexity and topological structure in non-vector space settings, especially in hyperbolic-type groups and semigroups.
• To demonstrate that convex hulls in divisible monoids can be precompact, enabling semicontinuity and minimax results.

Proposed method

• Introduces a gauge functional on topological groups that generalizes the classical gauge in locally convex vector spaces, using the group operation and neighborhood basis at identity.
• Uses the gauge functional to prove separation theorems for disjoint convex sets in locally convex topological groups.
• Applies the separation results to derive a Krein-Milman-type theorem, showing that compact convex sets in locally convex topological groups are the closed convex hull of their extreme points.
• Adapts Fan’s minimax theorem from [BZ86] to topological monoids by assuming convexity, lower/upper semicontinuity, and precompactness of convex hulls.
• Employs the structure of p-semidivisible monoids to construct sequences of points approximating rational convex combinations, enabling the use of semicontinuity arguments.
• Uses the continuous parametrization of convex sets via maps like Λ: ℝ → X (e.g., in the positive hyperbolic group) to transfer convexity and continuity properties from ℝ to the group.

Experimental results

Research questions

• RQ1 Can separation theorems in locally convex topological vector spaces be generalized to topological groups?
• RQ2 Does a Krein-Milman theorem hold in locally convex topological groups, and what conditions ensure that compact convex sets are the closed convex hull of their extreme points?
• RQ3 Under what conditions does a minimax theorem hold in topological monoids, particularly when the underlying space is not a vector space?
• RQ4 How do divisibility properties (e.g., p-semidivisibility) affect the behavior of convex functions and the structure of convex hulls in monoids?
• RQ5 To what extent can convexity and continuity in non-linear groups (e.g., the positive hyperbolic group) be reduced to properties on ℝ via continuous parametrization?

Key findings

• A gauge functional is defined for topological groups that behaves analogously to the classical gauge in locally convex vector spaces, enabling separation theorems.
• In locally convex, T₁ topological groups, all singletons are convex and no non-zero element has finite order, implying the group is torsion-free and uniquely divisible in the limit.
• The Krein-Milman theorem holds for compact convex subsets of locally convex topological groups: such sets are the closed convex hull of their extreme points.
• For p-semidivisible topological monoids, if the convex hull of any two points is precompact, then convex functions are determined by their values on dense rational combinations.
• A minimax theorem is established for convex-concave-like functions on compact convex subsets of p-semidivisible topological monoids, provided the convex hulls are precompact and functions are semicontinuous.
• In the positive hyperbolic group (parametrized by Λ: ℝ → X), convexity and continuity of functions on ℝ lift to the group, and minimax equality holds under standard semicontinuity and compactness assumptions.