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.
A Sequel to the First Six Books of the Elements of Euclid Hodges,Figis, and Co, Dublin, 1886
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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.