Sufficient conditions for the minimality of biconcave functions

Abstract

This paper describes sufficient conditions under which a biconcave function B : S = { ( x , y ) ∈ R 2 : x − 2 ≤ y ≤ x + 2 } → R \mathcal {B}\colon \mathfrak {S}=\{ (x,y)\in \mathbb {R}^2\colon x-2\le y\le x+2 \}\to \mathbb {R} is minimal with respect to an obstacle L : S → [ − ∞ , + ∞ ) L\colon \mathfrak {S}\to [-\infty ,+\infty ) , that is, it is the pointwise minimal among all biconcave functions B : S → R B\colon \mathfrak {S}\to \mathbb {R} that satisfy the inequality B ≥ L B\ge L .