Plane R-curves and their steepest descent properties I

Abstract

La pubblicazione è resa disponibile sotto le norme e i termini della licenza di deposito, secondo quanto stabilito dalla Policy per l'accesso aperto dell'Università degli Studi di Firenze (https://www.sba.unifi.it/upload/policy-oa-2016-1.pdf)

Plane R-curves II

“…This is an extension to the planar R-curves of the so called angle estimate of [2], which plays a fundamental role in order to get the bound for the length of γ in different situations even in CAT(0)-spaces [12]. As a consequence, in the same way as in [1], if γ is contained in a smaller disk, a bound of its length is obtained (Theorem 5.5). Moreover if γ is contained in a disk of arbitrary radius τ , bounds for the detour of γ and its length, depending on R and τ , are proved (Theorem 5.6) as in [1].…”

“…Let R > 0. Let Γ R be the class of the plane oriented local rectifiable curves γ satisfying the following property: for every x ∈ γ, let γx be the part of γ between the starting point and x and almost everywhere let t be the tangent vector to γ at x, then γx is not contained in the open circle centered at x + Rt and of radius R. These curves have been studied in [1] and have been called R-curves.…”

“…The steepest descent curves of quasi convex functions are curves of Γ [2,4,9]; the interest in the R-curves is that they are the steepest descent lines of functions whose level sets have reach greater than R, see [1];…”

“…On the other hand, the curves γ ∈ Γ R , with length parameter s, satisfy the following two properties [1]:…”

“…These properties are a generalization of 1 and of 2, respectively. In [1] a priori bounds for the length and the detour of a bounded γ ∈ Γ R have been proved under the assumption that γ is a C 1 curve. In this work the same bounds are obtained if one assumes that γ is merely rectifiable so the defining property holds a.e., only; first natural or easily obtained properties for C 1 R-curves are extended to arbitrary rectifiable plane R-curves: Theorem 4.5 and Corollary 4.6.…”