PerimeterA

PerimeterA is a generalized perimeter functional used in geometry and analysis. It assigns to a measurable set E in the plane a nonnegative value P_A(E) that depends on a fixed positive-definite matrix A, encoding directional weighting of the boundary. If the boundary ∂E is sufficiently smooth and ν_E(x) denotes the outer unit normal at x on ∂E, then P_A(E) is defined by P_A(E) = ∫_∂E ||A ν_E(x)|| ds, where ds is the arclength measure. Intuitively, directions aligned with the principal axes of A are penalized differently, so the perimeter cost is anisotropic.

When A is the identity matrix, P_A(E) reduces to the standard Euclidean perimeter. More generally, P_A corresponds to the anisotropic perimeter associated with the norm φ_A(v) = ||A v||, and it coincides with the anisotropic total variation of the indicator function of E. The functional is lower semicontinuous with respect to convergence in L^1 and is well-suited for variational problems.

The geometry of PerimeterA is governed by the Wulff shape associated with φ_A, namely the unit ball {v : φ_A(v) ≤ 1}. This shape plays a central role in minimizing P_A under volume constraints, leading to optimal sets whose boundaries approximate the Wulff shape. PerimeterA generalizes to higher dimensions with the boundary measure replaced by the appropriate surface measure.

Applications appear in image processing for edge-aware regularization and in materials science for modeling crystal growth and anisotropic surface energies. It provides a flexible framework to favor certain boundary orientations and to study shape optimization under directional penalties.

See also anisotropic perimeter, total variation, Wulff shape, BV.