On discretization of some extremal problems

TL;DR

This paper addresses two extremal problems involving monotone functions, transforming continuous problems into discrete ones to better understand the structure of extremal functions and derive their solutions.

Contribution

It introduces a method to discretize continuous extremal problems on monotone functions, revealing the structure of extremal functions and solving two specific problems.

Findings

Discrete problems are interesting and insightful.

Structural understanding of extremal functions.

Solutions to the two extremal problems.

Abstract

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class of functions; in the second one we find extremal values for the expectation of a random process with monotone trajectories at a random time. In both cases we reduce the continuous problems to their discrete counterparts. The obtained discrete problems are on the one hand interesting on their own, and on the other hand give a natural explanation of the structure of the extremal functions for the continuous problems.