Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convexity of sets in metric Abelian groups   In the present paper, we introduce a new concept of convexity which is
generated by a family of endomorphisms of an Abelian group. In Abelian groups
equipped with a translation invariant metric, we define the boundedness, the
norm, the measure of injectivity and the spectral radius of endomorphisms.
Beyond the investigation of their properties, our first main goal is an
extension of the celebrated R\aa dstr\"om Cancellation Theorem. Another result
generalizes the Neumann Invertibility Theorem. Next we define the convexity of
sets with respect to a family of endomorphisms and we describe the
set-theoretical and algebraic structure of the class of such sets. Given a
subset, we also consider the family of endomorphisms that make this subset
convex and we establish the basic properties of this family. Our first main
result establishes conditions which imply midpoint convexity. The next main
result, using our extension of the R\aa dstr\"om Cancellation Theorem, presents
further structural properties of the family of endomorphisms that make a subset
convex.

19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph