Diameter

The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of ${\displaystyle n}$-dimensional (convex or non-convex) object, such as a hypercube or a set of scattered points. The diameter or metric diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if ${\displaystyle S}$ is the subset and if ${\displaystyle \rho }$ is the metric, the diameter is

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

If the metric ${\displaystyle \rho }$ is viewed here as having codomain ${\displaystyle \mathbb {R} }$ (the set of all real numbers), this implies that the diameter of the empty set (the case ${\displaystyle S=\varnothing }$) equals ${\displaystyle -\infty }$ (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter of ${\displaystyle 0,}$[4] which corresponds to taking the codomain of ${\displaystyle d}$ to be the set of nonnegative reals.

For any solid object or set of scattered points in ${\displaystyle n}$-dimensional Euclidean space, the diameter of the object or set is the same as the diameter of its convex hull. In medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object.

In differential geometry, the diameter is an important global Riemannian invariant.

In planar geometry, a diameter of a conic section is typically defined as any chord which passes through the conic's centre; such diameters are not necessarily of uniform length, except in the case of the circle, which has eccentricity ${\displaystyle e=0.}$

Sign ⌀ in a technical drawing

Sign U+2205 ∅ EMPTY SET from an AutoCAD drawing in dim.shx font with an angle 16°. This font does not contain U+2300 ⌀ DIAMETER SIGN.

The symbol or variable for diameter, ⌀, is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. For example, photographic filter thread sizes are often denoted in this way.

In German, the diameter symbol (German Durchmesserzeichen) is also used as an average symbol (Durchschnittszeichen).

The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric. Jung's theorem provides more general inequalities relating the diameter to the radius.