Convex analysis in groups and semigroups: a sampler

Abstract

Abstract. We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only vector spaces. Some examples and counter-examples are also discussed.

A Perron–Frobenius type result for integer maps and applications

“…. It is known that a minimum of affine maps is concave (see [BG18]). As an immediate corollary of Theorem 3.2, we obtain the following results for a 'zigzag' map, that is, a map which is a maximum of minima of a finite number of affine maps.…”

“…Since the fixed point of the extended map B may be a non-integer point, we only have an 'approximate' eigenvector, which is suitably characterized by inequalities. A notion of concavity, originally introduced for groups in [BG18], is used to study discrete, concave maps. This paper is organized as follows.…”

Some remarks on convex analysis in topological groups

“…Similarly, a topological monoid is a monoid (i.e., commutative semigroup with unit), which is also a topological space, such that the addition operation is continuous. In [BG15] the present authors proposed a natural convexity structure for groups and monoids that coincides with the classical notion when the underlying structure is a vector space. It is then natural to ask when known algebraic or topological results for vector spaces still hold true in a group or monoid.…”

“…(1.1) and observes that for subadditive functions, the monoid provides the appropriate level of generality wherein to study infimal convolution. Our own motivation is discussed in [BG15] where also various illustrative examples are given and which provided a variety of primarily algebraic results.…”

“…While the notion of convexity is usually studied in the context of vector spaces, it can be defined and studied in a very general setting. We refer the reader to [vdV93] for more about abstract convexity, and to [BG15] for more about convexity in groups and monoids. In particular, given a space with an abstract collection of convex sets, we can define the following notion.…”