A set in Euclidean space R^d is convex set if it contains all the line segments connecting any pair of its points. If the set does not contain all the line segments, it is called concave.

A convex set is always star convex, implying pathwise-connected, which in turn implies connected.

A region can be tested for convexity in the Wolfram Language using the function Region`ConvexRegionQ[reg].

See also

Connected Set, Convex Domain, Convex Function, Convex Hull, Convex Optimization Theory, Convex Polygon, Convex Polyhedron, Convex Set, Delaunay Triangulation, Minkowski Convex Body Theorem, Simply Connected

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Benson, R. V. Euclidean Geometry and Convexity. New York: McGraw-Hill, 1966.Busemann, H. Convex Surfaces. New York: Interscience, 1958.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Convexity." Ch. A in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 6-47, 1994.Eggleston, H. G. Problems in Euclidean Space: Applications of Convexity. New York: Pergamon Press, 1957.Gruber, P. M. "Seven Small Pearls from Convexity." Math. Intell. 5, 16-19, 1983.Gruber, P. M. "Aspects of Convexity and Its Applications." Expos. Math. 2, 47-83, 1984.Guggenheimer, H. Applicable Geometry--Global and Local Convexity. New York: Krieger, 1977.Kelly, P. J. and Weiss, M. L. Geometry and Convexity: A Study of Mathematical Methods. New York: Wiley, 1979.Webster, R. Convexity. Oxford, England: Oxford University Press, 1995.

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Cite this as:

Weisstein, Eric W. "Convex." From MathWorld--A Wolfram Web Resource.

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