Diameter of a set

In mathematics, the diameter of a set of points in a metric space is the largest distance between points in the set. As an important special case, the diameter of a metric space is the largest distance between any two points in the space. This generalizes the diameter of a circle, the largest distance between two points on the circle. This usage of diameter also occurs in medical terminology concerning a lesion or in geology concerning a rock.

A bounded set is a set whose diameter is finite. Within a bounded set, all distances are at most the diameter.

Formal definition

The diameter of an object is the least upper bound (denoted "sup") of the set of all distances between pairs of points in the object.

Explicitly, if

S

{\displaystyle S}

is a set of points with distances measured by a metric

ρ

{\displaystyle \rho }

, the diameter is

diam

⁡

(

S

)

=

sup

x

,

y

∈

S

ρ

(

x

,

y

)

.

{\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).}

Of the empty set

The diameter of the empty set is a matter of convention. It can be defined to be zero,

−

∞

{\displaystyle -\infty }

, or undefined.