Discrete Midpoint Convexity
Satoko Moriguchi, Kazuo Murota, Akihisa Tamura, and Fabio Tardella

TL;DR
This paper introduces new classes of discrete convex functions based on midpoint convexity concepts on integer lattices, expanding the understanding of discrete convexity and its stability properties.
Contribution
It defines local and global discrete midpoint convex functions, positioning them between Lonvex and integrally convex functions, and proves their stability and proximity properties.
Findings
New classes of discrete convex functions are characterized.
These classes are stable under scaling and addition.
A proximity theorem similar to Lonvex functions is established.
Abstract
For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer lattice, we consider the analogous notion of discrete midpoint convexity, a discrete version of midpoint convexity where the value of the function at the (possibly noninteger) midpoint is replaced by the average of the function values at the integer round-up and round-down of the midpoint. It is known that discrete midpoint convexity on all line segments with integer endpoints characterizes L-convexity, and that it characterizes submodularity if we restrict the endpoints of the line…
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Taxonomy
TopicsOptimization and Variational Analysis · Point processes and geometric inequalities · Advanced Optimization Algorithms Research
