It is conjectured that any convex body in 
-dimensional Euclidean space
has an interior point lying on normals through 
distinct boundary points (Croft et al. 1991). This
has been proved for 
and 3 by Heil (1979ab, 1985). It is known that higher dimensions
always contain at least a 6-normal point, but the general conjecture remains open.
Concurrent Normals Conjecture
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References
Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51-79, 1967.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Concurrent Normals." §A3 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 14-15, 1991.Heil, E. "Existenz eines 6-Normalenpunktes in einem konvexen Körper." Arch. Math. (Basel) 32, 412-416, 1979a.Heil, E. "Correction to 'Existenz eines 6-Normalenpunktes in einem konvexen Körper.' " Arch. Math. (Basel) 33, 496, 1979b.Heil, E. "Concurrent Normals and Critical Points under Weak Smoothness Assumptions." Ann. New York Acad. Sci. 440, 170-178, 1985.Referenced on Wolfram|Alpha
Concurrent Normals ConjectureCite this as:
Weisstein, Eric W. "Concurrent Normals Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConcurrentNormalsConjecture.html