asdiameters

Asdiameters are a class of diameter-like measurements used to assess the spatial extent of a set under anisotropic, or direction-dependent, metrics. They generalize the conventional Euclidean diameter by incorporating directional weighting through a metric tensor, enabling the capture of elongated or oriented shapes that fair poorly under isotropic distance.

Formally, let S be a subset of Euclidean space R^n and let A be a symmetric positive definite matrix that defines an anisotropic metric d_A(x, y) = sqrt((x - y)^T A (x - y)). The as-diameter of S with respect to A is defined as D_A(S) = sup{ d_A(x, y) : x, y in S }. If A = I, the identity matrix, this reduces to the standard Euclidean diameter. For finite sets, the as-diameter is attained by a pair of points, though in continuous sets it is an extremal value.

Key properties include monotonicity with respect to set inclusion (larger sets do not have smaller as-diameters), and homogeneity under scaling: D_A(cS) = |c| D_A(S) for any scalar c. The value also reflects the orientation encoded by A; rotating the space relative to A alters the computed diameter unless A commutes with the rotation. When A emphasizes one axis over others, the as-diameter tends to be governed by the spread along the favored directions.

Variants and choices of A allow customization for applications such as shape analysis, computer vision, and robotics, where directional costs, uncertainties, or measurement biases are important. In practice, A is chosen to reflect the anisotropy of the problem domain, with D_A(S) offering a concise scalar summary of a set’s anisotropic extent.

See also: diameter, anisotropic metrics, norm-induced diameters, metric tensors.