Margarita Spirova

Minimal enclosing discs, circumcircles, and circumcenters in normed planes (Part I)

It is surprising that there are almost no results on the precise location of (all) minimal enclosing balls, circumballs, and circumcenters of simplices in finite-dimensional real Banach spaces. In this paper and a subsequent second part of it we give the starting point in this direction, also for computational investigations. More precisely, we present the first thorough study of these topics for triangles in arbitrary normed planes. In the present Part I we lay special emphasize on a complete description of possible locations of the circumcenters, and as a needed tool we give also a modernized classification of all possible shapes of the intersection that two homothetic norm circles can create. Based on this, we give in Part II the complete solution of the strongly related subject to find all minimal enclosing discs of triangles in arbitrary normed planes.

The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach

We give a discrete geometric (differential-free) proof of the theorem underlying the solution of the well known Fermat–Torricelli problem, referring to the unique point having minimal distance sum to a given finite set of non-collinear points in d-dimensional space. Further on, we extend this problem to the case that one of the given points is replaced by an affine flat, and we give also a partial result for the case where all given points are replaced by affine flats (of various dimensions), with illustrative applications of these theorems.

Covering Discs in Minkowski Planes

We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by k unit circles. In particular, we study the cases k = 3, k = 4, and k = 7. For k = 3 and k = 4, the diameters under con- sideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essentialrole in our proofs, namely Minkowskian bisectors, d-segments, and the monotonicity lemma.

Semi-Inner Products and the Concept of Semi-Polarity

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determining a Hilbert space. We use it on a finite dimensional real Banach space

On regular 4-coverings and their application for lattice coverings in normed planes

{({\mathbb{X}, \| \cdot \|})}

A new line associated with the triangle

(X,‖·‖) to define and investigate three concepts. First, we generalize that of antinorms, already defined in Minkowski planes, for even dimensional spaces. Second, we introduce normality maps, which in turn leads us to the study of semi-polarity, a variant of the notion of polarity, which makes use of the underlying semi-inner product.

Geometric algebra of strictly convex Minkowski planes

We introduce and study t-coverings in En, i.e., arrangements of proper translates of a convex body K ⊂ En sufficient to cover K. First, we investigate relations between t-coverings of the whole of K and t-coverings of its boundary only. Refining the notion of t-covering in several ways, we then derive, particularly for centrally symmetric convex bodies and n = 2, theorems which are interesting for the geometry of normed planes. These statements are related to respective generalizations of Tiţeica’s and Miquel’s theorem as well as to notions like Voronoi regions. We also compare t-coverings with coverings in the spirit of Hadwiger, using smaller homothetical copies of K instead of proper translates. This is done via a slight modification of Boltyanski’s and Hadwiger’s notion of illumination. Finally, we give upper bounds on the cardinalities of t-coverings.

Propellers in Affine Cayley-Klein Planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquels six-circles theorem from the Euclidean plane to all strictly convex normed planes.