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." In Discrete Geometry and Convexity (Ed. J. E. Goodman, E. Lutwak, J. Malkevitch, and R. Pollack). Ann. New York Acad. Sci. 440, pp. 170-178, 1985.Referenced on Wolfram|Alpha
Concurrent Normals ConjectureCite this as:
Weisstein, Eric W. "Concurrent Normals Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConcurrentNormalsConjecture.html