Fenton type minimax problems for sum of translates functions

TL;DR

This paper extends Fenton's minimax problem analysis for sum of translates functions by removing previous assumptions, establishing the equality of minimax and maximin values, and characterizing extremal configurations as equioscillation points.

Contribution

It generalizes existing results by eliminating assumptions on the kernel and field functions, providing a broader framework for concave kernel functions in minimax problems.

Findings

Proves the equality of minimax and maximin values.

Establishes existence and characterization of extremal configurations.

Generalizes previous results to broader class of kernel functions.

Abstract

Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded "field function", and $K:[-1,1]\to {\underline{\mathbb{R}}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, $\mathbf{x}:=(x_1,\ldots,x_n)$ with $0\le x_1\le\dots\le x_n\le 1$, and $\nu_1,\dots,\nu_n>0$ are fixed. We analyze the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j(\mathbf{x}):=\sup_{x_j\le t\le x_{j+1}} F(\mathbf{x},t)$, with $x_0:=0$, $x_{n+1}:=1$; and study the optimization (minimax and maximin) problems $\inf_{\mathbf{x}}\max_j m_j(\mathbf{x})$ and $\sup_{\mathbf{x}}\min_j m_j(\mathbf{x})$. The main result is the equality of these quantities, and provided $J$ is upper semicontinuous, the existence of extremal configurations and their description as equioscillation points $\mathbf{w}$. In our previous papers we obtained results for the case of singular kernels, i.e., when $K(0)=-\infty$ and the field $J$ was assumed to be upper semicontinuous. In this work we get rid of these assumptions and prove common generalizations of Fenton's and our previous results, arriving at the greatest generality in the setting of concave kernel functions.