For a convex body in R^n, a locally shortest chord through an interior point must have smooth endpoints whose supporting hyperplane normals meet on the plane perpendicular to the chord.
Coxeter , Jan van de Craats, Philon Lines in Non-Eucledean Planes , Journal of Geom- etry , Vol
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A Generalization of a Classical Geometric Extremum Problem
For a convex body in R^n, a locally shortest chord through an interior point must have smooth endpoints whose supporting hyperplane normals meet on the plane perpendicular to the chord.