Scaled Relative Graphs in Normed Spaces

Abstract

The paper extends the Scaled Relative Graph (SRG) framework of Ryu, Hannah, and Yin from Hilbert spaces to normed spaces. Our extension replaces the inner product with a regular pairing, whose asymmetry gives rise to directional angles and, in turn, directional SRGs. Directional SRGs are shown to provide geometric containment tests certifying key operator properties, including contraction and monotonicity. Calculus rules for SRGs under scaling, inversion, addition, and composition are also derived. The theory is illustrated by numerical examples, including a graphical analysis of the contractivity of Bellman operators using directional SRGs.

arXiv commentTo appear in the Proceedings of the 65th IEEE Conference on Decision and Control (CDC 2026)