Ellipses numbers and geometric measure representations

Abstract

Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A(r) = π(a,b)r2, the number π(a,b) = abπ reflects a generalized circumference property U(a,b)(r) = 2π(a,b)r of the ellipses E(a,b)(r) with main axes of lengths 2ra and 2rb, respectively. In this sense, the number π(a,b) is an ellipse number w.r.t. the Minkowski functional r of the reference set E(a,b)(1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.