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The smallest-circle problem or minimum covering circle problem is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. The corresponding problem in n-dimensional space, the smallest bounding-sphere problem, is to compute the smallest n-sphere that contains all of a given set of points.[1] The smallest-circle problem was initially proposed by the English mathematician James Joseph Sylvester in 1857.

The smallest-circle problem in the plane is an example of a facility location problem (the 1-center problem) in which the location of a new facility must be chosen to provide service to a number of customers, minimizing the farthest distance that any customer must travel to reach the new facility. Both the smallest circle problem in the plane, and the smallest bounding sphere problem in any higher-dimensional space of bounded dimension, may be solved in linear time.

Most of the geometric approaches for the problem look for points that lie on the boundary of the minimum circle and are based on the following simple facts:

The minimum covering circle is unique.

The minimum covering circle of a set S can be determined by at most three points in S which lie on the boundary of the circle. If it is determined by only two points, then the line segment joining those two points must be a diameter of the minimum circle. If it is determined by three points, then the triangle consisting of those three points is not obtuse.