Non-linear quotient mappings of the plane and inscribed equilateral polygons in centrally symmetric convex bodies

Hutchins, Ricky (2024). Non-linear quotient mappings of the plane and inscribed equilateral polygons in centrally symmetric convex bodies. University of Birmingham. Ph.D.

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Abstract

In the present work, we are concerned with two underlying topics, which at first seem inherently disjoint, but are connected in a non-obvious way.

The first topic we study is the behaviour of non-linear quotient mappings whose domain is the plane; we pay particular attention to Lipschitz quotient mappings of the plane. This forms the underlying material in Chapters 2 and 3.

In Chapter 2 we correct a mistake from Johnson et al. Namely, we give a valid proof of the statement that for a fixed complex polynomial $P$ in one complex variable there exists a homeomorphism of the plane $h$ such that $P\circ h$ is a Lipschitz quotient mapping of the plane. Further, we introduce the notion of strong co-Lipschitzness, and prove the logical equivalence between the long standing conjecture that all Lipschitz quotient mappings from $\mathbb{R}^n$ to itself are discrete and the necessity for every Lipschitz quotient mapping from $\mathbb{R}^n$ to itself, $n\geq 3$ , to be strongly co-Lipschitz.

Chapter 3 is dedicated to improving the lower estimates for the ratio of constants $L/c$ for any $2$ -fold planar Lipschitz quotient mappings in polygonal norms.

Chapters 4 and 5 concern the existence of inscribed equilateral polygons in centrally symmetric convex bodies. We investigate the extremal inscribed equilateral polygons and determine, for a large class of norms, when such polygons are essentially equivalent.

Finally, we consider the level sets of uniformly continuous, co-Lipschitz mappings defined on the plane; this forms Chapter 6. We show for any uniformly continuous, co-Lipschitz mapping $f:(\mathbb{C},{\|\cdot\|})\to\mathbb{R}$ , where $\|\cdot\|$ is any norm of the plane, that the maximal number of components $n(f)$ of the level sets of $f$ is intimately related to weak Lipschitz and co-Lipschitz constants of $f$ as well as the maximal possible edge length, in terms of $\|\cdot\|$ , over all inscribed equilateral polygons. Further, we obtain a sharp bound for a certain class of norms $\|\cdot\|$ which possess a particular separation property.