On the Optimality of Piecewise Linear Max-norm Enclosures based on Slefes

Subdividable linear efficient function enclosures (Slefes) provide, at low cost, a piecewise linear pair of upper and lower bounds f + , f − , that sandwich a function f on a given interval: f + � ff − . In practice, these bounds are observed to be very tight. This paper addresses the question just how close to optimal, in the max-norm, the slefe construction actually is. Specifically, we compare the width f + f − of the slefe to the narrowest possible piecewise linear enclosure of f when f is a univariate cubic polynomial.