Related papers: Critical loci of convex domains in the plane

For a compact convex subset $K $ of a locally convex Hausdorff space, a measurement on $A(K)$ is a finite family of positive elements in $A(K)$ normalized to the unit constant $1_K$, where $A(K)$ denotes the set of continuous real affine…

The Real Projective Plane is the lowest dimensional orbifold which, when combined with the usual Minkowski space-time, gives rise to a unique model in six flat dimensions possessing an exact Kaluza Klein (KK) parity as a relic symmetry of…

Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be…

For Classical compact Lie groups, we use Macdonald's formula \cite{Ma} and Ricci curvature for analyzing a "concentration locus", which is a tool to detect where a sequence of metric, Borel measurable spaces concentrates its measure.

We argue that topological compactons (solitons with compact support) may be quite common objects if $k$-fields, i.e., fields with nonstandard kinetic term, are considered, by showing that even for models with well-behaved potentials the…

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any…

In this paper the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in CR(Y) where Y is a compact Hausdorff space. Then a…

We describe the topology of critical loci of coamoeba of generic affine planes in four-space.

In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By…

Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove…

We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant $1$.…

For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determined all Hausdorff limit domains that one can obtain…

Let K be a complete, algebraically closed, nonarchimedean valued field, and let f(z) be a rational function in K(z) of degree d at least 2. We show there is a natural way to assign non-negative integer weights w_f(P) to points of the…

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to…

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…

We give a topological model of the critical locus for complex H\'{e}non maps that are perturbations of the quadratic polynomial with disconnected Julia set.

For a $(-1)$-shifted Lagrangian in a critical locus, we construct a homomorphism from the $K$-group of matrix factorisations of the critical locus to the $K$-group of the Lagrangian, partially answering the Joyce-Safronov conjecture. The…

Quantum gravity is investigated in the limit of a large number of space-time dimensions, using as an ultraviolet regularization the simplicial lattice path integral formulation. In the weak field limit the appropriate expansion parameter is…

We use bicombings on arcwise connected metric spaces to give definitions of convex sets and extremal points. These notions coincide with the customary ones in the classes of normed vector spaces and geodesic metric spaces which are convex…

Let $\beta_1,...,\beta_n$ be distinct points in the open unit disc in the complex plane, none of which is the origin, and let $H^1$ be the Hardy space. Define a closed convex set in $\mathbb{C}^{n}$ by $\Lambda = \{…