autovariograms

Autovariogram is a function used in geometry and geometric probability that measures how much a set overlaps with itself after a translation. For a measurable set A in Euclidean space R^n with finite volume, the autovariogram g_A is defined by g_A(h) = vol(A ∩ (A + h)), where h ∈ R^n and A + h denotes the translate of A by h. Equivalently, g_A is the autocorrelation of the indicator function 1_A, since g_A(h) = ∫ 1_A(x) 1_A(x + h) dx.

The autovariogram has several basic properties. It is even: g_A(h) = g_A(-h), and it attains its maximum at h = 0, where g_A(0) = vol(A). The support of g_A lies in the difference set A − A, since if h is outside A − A the overlap is empty. In many settings g_A is continuous (and even smooth for smooth boundaries) and can be interpreted as the Fourier transform squared of 1_A, giving a positive-definite, real-valued function.

The study of autovariograms is closely related to the covariogram problem. The covariogram (or autovariogram) problem asks whether the function g_A determines the shape of A up to translations and reflections. For many classes of convex bodies in the plane and in some higher-dimensional cases, the covariogram does determine the body up to those symmetries, though there are known non-uniqueness results in higher dimensions and for certain non-convex sets.

Applications of autovariograms include shape reconstruction from overlap data, materials science and imaging to characterize microstructures, and stochastic geometry where covariograms extend to random sets. They also provide a bridge to autocorrelation concepts in signal processing and spatial statistics.