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Geometric Properties of Blaschke-Like Maps on Domains with a Conic Boundary
For a circle C contained in the unit disk, the necessary and sufficient condition for the existence of a triangle inscribed in the unit circle and...
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Blaschke Products, Level Sets, and Crouzeix’s Conjecture
We study several problems motivated by Crouzeix’s conjecture, which we consider in the special setting of model spaces and compressions of the shift...
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The Beauty of Blaschke Products
This chapter is dedicated to showing how visual tools, using geometry and color, can be used to enhance the understanding of statements about complex...
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The Beauty of Blaschke Products
This chapter is dedicated to showing how visual tools, using geometry and color, can be used to enhance the understanding of statements about complex...
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Isoperiodic families of Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal pencil
Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range...
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Reconstructing Planar Ellipses from Translation-Invariant Minkowski Tensors of Rank Two
Minkowski tensors contain information about shape and orientation of the underlying convex body. We make this precise by showing that reconstructing...
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Invariant Center Power and Elliptic Loci of Poncelet Triangles
We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their...
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Poncelet triangles: a theory for locus ellipticity
We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families...
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Four equivalent properties of integrable billiards
By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and...
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Points in the plane, lines in space
The 1911 Grünwald–Blaschke map** is reviewed from the point of view of a particular Clifford algebra. This is a map** between the group of proper...
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Theorems from the Theory of Convex Bodies
After the two well-known theorems discussed in the first two sections of this chapter some more specialized theorems will follow, mainly concerning...
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Numerical bounds on the Crouzeix ratio for a class of matrices
We provide numerical bounds on the Crouzeix ratio for KMS matrices A which have a line segment on the boundary of the numerical range. The Crouzeix...
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Nevanlinna Domains and Uniform Approximation by Polyanalytic Polynomial Modules
We consider the concept of a Nevanlinna domain and its modifications: the concept of a locally Nevanlinna domain and the concept of g-Nevanlinna...
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On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs
Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC)...
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Basic Results of Integral Geometry
The results in this chapter are of a geometric nature, and they emanate from the basic concept of motion-invariant measure associated with a mobile...
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On Foci of Ellipses Inscribed in Cyclic Polygons
Given a natural number n ≥ 3 and two points a and b in the unit disk...
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Symmetrizations
In this chapter we present the operation of symmetrization of sets and briefly touch on the strictly connected operation of rearrangement of...
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Some Theorems from Elementary Geometry
In this chapter we compile the necessary preliminaries from elementary geometry. They are mainly about some well-known concepts and theorems,...
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Efficiency of Packings and Coverings with a Sequence of Convex Disks
We search for those convex disks which are: 1) the least efficient for packing the plane, 2) the least efficient for covering it. Thus, in a certain...
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Peabodies of constant width
We describe a new family of 3-dimensional bodies of constant width that we have called peabodies, obtained from the Reuleaux tetrahedron by replacing...
