On steepest descent curves for quasi convex families in $R^n$

Marco Longinetti; Paolo Manselli; Adriana Venturi

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arXiv · 1303.3721

On steepest descent curves for quasi convex families in $R^n$

Marco Longinetti·Paolo Manselli·Adriana Venturi

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Abstract

A connected, linearly ordered path $\ga \subset R^n$ satisfying $$ x_1\prec x_2\prec x_3 \in \ga, and x_1 \prec x_2 \prec x_3 \Rightarrow |x_2 - x_1| \leq | x_3 - x_1|$$ is shown to be a rectifiable curve; a priori bounds for its length are given; moreover, these paths are generalized steepest descent curves of suitable quasi convex functions. Properties of quasi convex families are considered; special curves related to quasi convex families are defined and studied; they are generalizations of steepest descent curves for quasi convex functions and satisfy the previous property. Existence, uniqueness, stability results and length's bounds are proved for them.

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Marco Longinetti, Paolo Manselli, Adriana Venturi. 2013-03-15. On steepest descent curves for quasi convex families in $R^n$. https://arxiv.org/abs/1303.3721

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