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Concurrent Normals Conjecture

It is conjectured that any convex body in n-dimensional Euclidean space has an interior point lying on normals through 2n distinct boundary points (Croft et al. 1991). This has been proved for n=2 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.

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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.

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Concurrent Normals Conjecture

Cite this as:

Weisstein, Eric W. "Concurrent Normals Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConcurrentNormalsConjecture.html

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