Diameter - Generalizations


The four definitions given above are special cases of a more general definition. The diameter of a subset of a metric space is the least upper bound of the distances between pairs of points in the subset. So, if A is the subset, the diameter is

sup { d(x, y) | x, yA } .

If the distance function d is viewed here as having codomain R (the set of all real numbers), this implies that the diameter of the empty set (the case A = ∅) equals −∞ (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter equal to 0, which corresponds to taking the codomain of d to be the set of nonnegative reals.

In differential geometry, the diameter is an important global Riemannian invariant. In plane and coordinate geometry, a diameter of a conic section is 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 e = 0.

In medical parlance the diameter of a lesion is the longest line segment whose endpoints are within the lesion.

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