Same average in every direction
Imre B\'ar\'any, G\'abor Domokos

TL;DR
This paper investigates the conditions under which the average number of vertices of plane intersections with a polytope remains constant regardless of the direction of the intersecting plane.
Contribution
It characterizes polytopes for which the average number of vertices in plane intersections is direction-independent.
Findings
Identifies polytopes with constant average vertices across all directions.
Provides criteria for polytopes with uniform intersection properties.
Explores geometric conditions for directional invariance.
Abstract
Given a polytope and a non-zero vector , the plane intersects in convex polygon for where and , is the scalar product of . Let denote the average number of vertices of on the interval . For what polytopes is a constant independent of ?
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Taxonomy
TopicsPoint processes and geometric inequalities · Computational Geometry and Mesh Generation · Geometric Analysis and Curvature Flows
