In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical H...

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Let ν stand for the homogeneous dimension of a Carnot group and let m be the index of the last vector space of the corresponding Lie algebra. We prove that the (ν −m− 1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimat...

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen’s S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

In this article we describe recent (within the past 4 or 5 years) results in minimal surface theory, with particular emphasis on the existence and regularity theory. For a more general survey, including some technical background discussion and a more comprehensive bibliography, the reader is referred to [30]. Throughout the article «?f will denote ÄJ-dimensional Hausdorff measure (in Euclidean ...

In the present paper we sketch the proof of the fact that for any open connected set Ω ⊂ Rn+1, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable.

If X is a subset of R , let Hk(X) denote the k-dimensional hausdorff measure of X . We write Ik(X) = 0 if H(πKX) = 0 for almost every k-plane K in R , where πK : R → K is orthogonal projection. Otherwise we write Ik(X) > 0. This paper gives a proof of the following theorem. 1.1. Structure Theorem. Let X be a set in R with Hk(X) < ∞ and Ik(X) > 0. Then there is a k-dimensional C submanifold M wi...

This paper proposes an oriented Hausdorff similarity measure for object matching. The oriented Hausdorff distance (HD) measure is introduced by replacing the distance concept of conventional HD algorithms by a similarity measure. The orientation at each ~ i x e l is used for removing incorrect correspondences. Various experiments show that the perFigure 1: Block diagram for the Hausdorff distan...

Let (Ω, d) be a metric space where Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension and let B be a partition of Ω. The coherent upper conditional prevision defined as the Choquet integral with respect to its associated Hausdorff outer measure is proven to satisfy the disintegration property and the conglomerative principle on every partition.

We have collected definitions and basic results for the (centered ball) density in metric space with respect to an arbitrary Hausdorff function. We have kept the definitions general: we do not assume the Hausdorff functions are continuous or blanketed, and we do not assume the metric space is a subset of Euclidean space. We discuss the covering measure (= centered Hausdorff measure) and packing...