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.
Lhuilier Principiorum calculi differentialis et integralis expositio elementaris, Tubingae,1795, p
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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.