A Universal Boundary--Bulk Scaling Functional

Key Points

• This research aims to quantify the competitive relationships between boundary and bulk structures using a new morphometric functional.
• Introduced a new morphometric functional 'n' to quantify boundary dominance per effective degree of freedom.
• Developed a scaling theorem to characterize asymptotic behaviors of the morphometric functional.
• Analyzed distinct limiting behaviors of classical geometric structures under the proposed framework.
• Identified distinct limiting behaviors of classical geometric structures as determined by the morphometric functional.
• Demonstrated an extremal ordering between isotropic and axis-aligned geometries.
• Validated the scaling theorem across various systems including convex geometry and fractal systems.