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Corpus ID: 208248175

On Convexity, Mid-Point Convexity and Hausdorff Measures of Sets.

@article{Guo2019OnCM,
title={On Convexity, Mid-Point Convexity and Hausdorff Measures of Sets.},
author={Shaoming Guo and Tian Lan and Yakun Xi},
journal={arXiv: Classical Analysis and ODEs},
year={2019}
}

We give a complete characterization of the size of Borel sets that are mid-point convex but not (essentially) convex, in terms of their Hausdorff dimensions and Hausdorff measures.

Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.Expand

Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as
$$kA =\{ {a}_{1} + \cdots + {a}_{k} : {a}_{i} \in A\}.$$
We show that for any… Expand

Central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem.Expand