Abstract. We prove analogs of classical almost sure dimension theorems for Euclidean projection mappings in the first Heisenberg group, equipped with a sub-Riemannian metric.
Abstract. It is hereby established that, in Euclidean spaces of finite dimension, bounded selfcontracted curves have finite length. This extends the main result of [6] concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof borrows heavily from a geometric idea of [13] employed for the study of regular enough curves, and can be seen as a nonsmooth adaptation of the latter, albeit a nontrivial one. Applications to continuous and discrete dynamical systems are discussed: continuous self-contracted curves appear as generalized solutions of nonsmooth convex foliation systems, recovering a hidden regularity after reparameterization, as consequence of our main result. In the discrete case, proximal sequences (obtained through implicit discretization of a gradient system) give rise to polygonal self-contracted curves. This yields a straightforward proof for the convergence of the exact proximal algorithm, under any choice of parameters.
For a metric space X, we study the space D ∞ (X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D ∞ (X) is compared with the space LIP ∞ (X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D ∞ (X) with the Newtonian-Sobolev space N 1,∞ (X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D ∞ (X) = N 1,∞ (X).
Abstract. In this work we explore the preservation of quasiconvexity and ∞-Poincaré inequality under sphericalization and flattening in the metric setting. The results developed in [22] show the preservation of Ahlfors regularity, doubling property and the p-Poincaré inequality for 1 ≤ p < ∞ under the sphericalization and flattening transformations provided the underlying metric space is annularly quasicovex. In this work, we propose a weaker assumption to still preserve quasiconvexity and ∞-Poincaré inequality, called radially star-like quasiconvexity (corresponding to sphericalization) and meridian-like quasiconvexity (corresponding to flattening) extending in particular a result in [8] to a wider class of metric spaces and covering the case p = ∞ in [22].
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