Convex

A set in Euclidean space  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].

REFERENCES:

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.