Multivariate mode hunting is of increasing practical importance. Only a few such methods exist, however, and there usually is a trade off between practical feasibility and theoretical justification. In this paper we attempt to do both. We propose a method for locating isolated modes (or better, modal regions) in a multivariate data set without pre-specifying their total number. Information on s...

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve γ : R → R , N ≥ 2, is locally 1-monotone.

Let γ(E) be the analytic capacity of a compact set E and let γ+(E) be the capacity of E originated by Cauchy transforms of positive measures. In this paper we prove that γ(E) ≈ γ+(E) with estimates independent of E. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that γ is semiadditive.

Let γ(E) be the analytic capacity of a compact set E and let γ+(E) be the capacity of E originated by Cauchy transforms of positive measures. In this paper we prove that γ(E) ≈ γ+(E) with estimates independent of E. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that γ is semiadditive.