diametersurface

Diametersurface is not a single, universally defined object in mathematics or science. In practice, the phrase is used informally to describe related surface-like constructs that connect the notion of a diameter with a geometric surface. Because the term is not standardized, its precise meaning depends on context. Two common interpretations appear in informal usage:

Diameter map surface. One interpretation views a diametersurface as the graph of a diameter-related function over a directional domain. For a compact set K in Euclidean space, one can consider, for each direction on the unit sphere, the maximum distance between boundary points in that direction (or the corresponding width). The resulting function, when plotted over directions, yields a surface in a higher-dimensional space that encodes how the diameter or width of K varies with direction. This is related to, but distinct from, the classic width or support function used in convex geometry. For simple shapes like a circle, this diameter map is constant; for ellipses or more irregular bodies, it varies with direction, producing a nontrivial surface.

Antipodal diameter surface. A second interpretation treats the set of all pairs of points in a shape that achieve the diameter. Formally, if diam(K) is the maximum distance between points of K, then the set of all pairs (x, y) in K × K with ||x − y|| = diam(K) can be considered a diameter surface. This object lives in a product space and may project to the original shape in various ways, producing a surface-like or curve-like collection depending on the geometry of K.

Examples and properties. For smooth, strictly convex bodies, the diameter is attained at antipodal boundary points in many directions, and the diameter map tends to be piecewise smooth with potential changes at transition directions. Computation of these surfaces often involves convex analysis and geometry of support points.

Applications. Diametersurface concepts appear in shape analysis, computer graphics, and collision detection, where understanding how diameter varies with direction or which point pairs realize the diameter informs symmetry, stability, and optimization tasks.

See also. Diameter, width function, support function, convex body, antipodal pairs.