Angular measures and Birkhoff orthogonality in Minkowski planes

Area of Science:

• * Functional Analysis
• * Geometric Measure Theory
• * Normed Plane Geometry

Background:

• * Birkhoff orthogonality is a geometric concept in normed spaces, generalizing orthogonality in Euclidean planes.
• * B-measures are specialized angular measures on the unit circle with specific properties related to Birkhoff orthogonality.
• * Previous work by Fankhänel introduced B-measures, motivating further investigation into their existence and properties.

Purpose of the Study:

• * To characterize normed planes that admit the existence of a B-measure.
• * To establish conditions on normed planes for the existence of this specialized angular measure.

Main Methods:

• * Definition of Birkhoff orthogonality for unit vectors in a normed plane.
• * Definition of a B-measure based on Birkhoff orthogonality and angular properties on the unit circle.
• * Analysis of geometric properties of normed planes related to the unit disc and supporting lines.

Main Results:

• * A characterization of normed planes admitting a B-measure is presented.
• * The study identifies specific geometric conditions required for a normed plane to possess a B-measure.

Conclusions:

• * The existence of B-measures is tied to the geometric structure of the normed plane.
• * This work provides a deeper understanding of the relationship between geometric properties and measure theory in normed spaces.