Interval Analysis and Calculus for Interval-Valued Functions of a Single Variable—Part II: Extremal Points, Convexity, Periodicity

Abstract

We continue the presentation of new results in the calculus for interval-valued functions of a single real variable. We start here with the results presented in part I of this paper, namely, a general setting of partial orders in the space of compact intervals (in midpoint-radius representation) and basic results on convergence and limits, continuity, gH-differentiability, and monotonicity. We define different types of (local) minimal and maximal points and develop the basic theory for their characterization. We then consider some interesting connections with applied geometry of curves and the convexity of interval-valued functions is introduced and analyzed in detail. Further, the periodicity of interval-valued functions is described and analyzed. Several examples and pictures accompany the presentation.

Keywords

• interval-valued gh-difference
• comparison index
• partial orders
• lattice of real intervals
• convex interval functions
• periodic interval functions