Diameterlike

Diameterlike is an informal term used in geometry and analysis to describe measures of a set’s spatial extent that generalize the usual notion of diameter. The classical diameter of a nonempty bounded subset S of a metric space (X,d) is diam(S) = sup{d(x,y): x,y in S}. Diameterlike quantities broaden this idea by incorporating center, direction, or weighting, yielding a family of related measures rather than a single value.

One common center-based approach defines, for S ⊆ X and a chosen center c in X, D(S,c) = sup{d(c,x): x in S}. This radial extent describes how far S reaches from the point c. The minimum of D(S,c) over all centers c is the radius of S, while the collection of values {D(S,c): c in X} characterizes the spread of S around different centers. This viewpoint emphasizes how a set sits with respect to potential centers, rather than how far apart its points are from each other.

A directional, or width-based, version is particularly common in Euclidean spaces. For a unit vector u, the directional width is W_S(u) = max_{x in S} ⟨u, x⟩ − min_{x in S} ⟨u, x⟩, where ⟨.,.⟩ denotes the inner product. The function u ↦ W_S(u) is a diameterlike profile that describes the extent of S in every direction. The overall diameter often satisfies diam(S) = max_{u unit} W_S(u), and the width function is invariant under translations and scales linearly with dilations.

Applications of diameterlike measures appear in shape analysis, computer vision, and geometric tomography, where a single diameter value may be insufficient to capture a shape’s geometry. See also diameter, radius, width, and the support function.

Note: diameterlike is not a standardized mathematical term; it is used informally to refer to these generalized, direction- or center-dependent extents.