Helly Family - The Helly Property

The Helly Property

If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k for which the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as the Helly property. A 2-Helly family is also known as a Helly family.

A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1) is called injective or hyperconvex. The existence of the tight span allows any metric space to be embedded isometrically into a space with Helly dimension 1.