Mathematics

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.

A convex polygon Q is inscribed in a convex polygon P if every side of P contains at least one vertex of Q . We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex n -gon in O(n) and O( n 3 ) time, respectively. We also investigate other variants of this problem.

We prove that if a convex set in Cn contains two inscribed complex ellipsoid of maximal volume then one is a translate of the other. On the other hand, the circumscribed complex elipsoid of minimal volume is unique. As application we prove the complex analoge of Brunn's characterization of ellipsods.

The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the relative Chebyshev radius. In particular, we determine the relative Chebyshev radius for an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and convex n-gons of a given relative Chebyshev radius.

The K -hull of a compact set A??R d , where K??R d is a fixed compact convex body, is the intersection of all translates of K that contain A . A set is called K -strongly convex if it coincides with its K -hull. We propose a general approach to the analysis of facial structure of K -strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of k -dimensional faces, for all k=0,??d?? . We then apply our theory in the case when A= ? n is a sample of n points picked uniformly at random from K . We show that in this case the set of x??R d such that x+K contains the sample ? n , upon multiplying by n , converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding f -vector of the K -hull of ? n to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the f -vector.

We study the dynamic geometry, loci, and invariants of three Poncelet families associated with three distinct concentric Ellipse pairs: with-incircle, with-circumcircle, and homothetic. Most of their properties run parallel to those of 3 well-studied families: elliptic billiard (confocal pair), Chapple's poristic triangles, and the Brocard porism, allowing us to organize them in three "similar" groups.

Kusner asked if n+1 points is the maximum number of points in R n such that the ℓ p distance between any two points is 1 . We present an improvement to the best known upper bound when p is large in terms of n , as well as a generalization of the bound to s -distance sets. We also study equilateral sets in the ℓ p sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when p=∞ , p is even, and for any 1≤p<∞ .

We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai--Komlós--Tusnády optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.

The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After a brief introduction, which gives the main theme, I present the main results, according to a synthetic view of the subject, rather that chronologically. First, I give some variation on the axiomatic treatment of projective geometry, followed by new results on quaternionian geometry, followed by results in geometry over the reals which are generalized over arbitrary fields, then those which depend on properties of finite fields. I then present results in finite mechanics. The role of the computer, which was essential for these inquiries, is briefly surveyed. The methodology to obtain illustrations by drawings is described. I end with a table which enumerates enclosed additional material.

We consider self-similar sets possessing finite intersection property and analyze topological structure nearby their local cut points.