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 Solution d’un probl` eme de g´ eom´ etrie, d´ ependant de la th´ eorie des maximis et min- imis, Annales de Math´ ematiques pures et appliqu´ ees, tome 2 (1811-1812), 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.