In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a ...

Given a compact metric space (Ω, d) equipped with a non-atomic, probability measure m and a real, positive decreasing function ψ we consider a ‘natural’ class of lim sup subsets Λ(ψ) of Ω. The classical lim sup sets of ‘well approximable’ numbers in the theory of metric Diophantine approximation fall within this class. We show that m(Λ(ψ)) > 0 under a ‘global ubiquity’ hypothesis and the diverg...

Let A be a compact set in Rp of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure μs is the unique Borel probability measure with support in A that minimizes Is(μ) := " 1 |x − y|s dμ(y)dμ(x) over all such probability measures. If A is strongly (Hd , d)-rectifiable, then μs converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s...

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Ra...

Let (Ω, d) be a metric space and let B be a partition of Ω. For every set B of B with positive and finite Hausdorff outer measure in its Hausdorff dimension, a coherent conditional measure of risk is defined as the Choquet integral with respect to Hausdorff outer measure. Two risks are defined to be s-independent if the atoms of the classes generated by their weak upper level sets are s-indepen...

One theme in noncommutative geometry is the recovery of classical geometric data from operator algebraic descriptions of various spaces, formalized through the concept of a spectral triple. This is defined below but, roughly speaking, consists of a C-algebra acting on a Hilbert space, giving co-ordinates for the space, and a Dirac operator, which encodes geometric information. As a prototypical...

Comparing face images using the Hausdorff distance is one of the face matching and fast screening techniques. As a fast screening technique, the computational efficiency is a key issue. An efficient method for Hausdorff distance-based face matching and fast screening is proposed. The method utilises dominant points, instead of edge maps, as features for measuring similarity. A new formulation o...