For a decreasing real valued function ψ, pair (A,b) of m×n matrix A and b∈Rm is said to be ψ-Dirichlet improvable if the system‖Aq+b−p‖m<ψ(T)and‖q‖n<T has solution p∈Zm, q∈Zn for all sufficiently large T, where ‖⋅‖ denotes supremum norm. Kleinbock Wadleigh (2019) established an integrability criterion Lebesgue measure non-improvable set. In this paper, we prove similar Hausdorff Also, extend re...

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conque...

The Falconer conjecture says that if a compact planar set has Hausdorff dimension > 1, then the Euclidean distance set ∆(E) = {|x − y| : x, y ∈ E} has positive Lebesgue measure. In this paper we prove, under the same assumptions, that for almost every ellipse K, ∆ K (E) = {||x − y|| K : x, y ∈ E} has positive Lebesgue measure, where || · || K is the norm induced by an ellipse K. Equivalently, w...

The Voronoi Covariance Measure of a compact set K of R is a tensor-valued measure that encodes geometrical information on K and which is known to be resilient to Hausdorff noise but sensitive to outliers. In this article, we generalize this notion to any distance-like function δ and define the δ-VCM. Combining the VCM with the distance to a measure and also with the witnessed-k-distance, we get...