Largest empty sphere

In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles.

The dashed circle is the outline of the largest empty sphere in the close-packing of spheres. See also Interstitial defect.

Finding the largest empty circle using the Voronoi diagram (two solutions).

Two dimensions

The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles.

A common special case is as follows. Given n points in the plane, find a largest circle centered within their convex hull and enclosing none of them. The problem may be solved using Voronoi diagrams in optimal time ${\displaystyle \Theta (n\,\log \,n)}$.^[1][2]

See also

• Bounding sphere
• Farthest-first traversal
• Largest empty rectangle

References

1. G. T. Toussaint, "Computing largest empty circles with location constraints," International Journal of Computer and Information Sciences, vol. 12, No. 5, October, 1983, pp. 347-358.
2. Megan Schuster, "The Largest Empty Circle Problem"