Related papers: C-transfinite diameter

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…

According to a classical result of Gr\"unbaum, the transversal number $\tau(\F)$ of any family $\F$ of pairwise-intersecting translates or homothets of a convex body $C$ in $\RR^d$ is bounded by a function of $d$. Denote by $\alpha(C)$…

Let $K$ be a compact convex body in $\mathbb R^n.$ For any affine line $L,$ denote $\widehat{\chi}_K(L)=\int_{L}\chi_K(x)dl(x),$ where $dl$ is the arc length measure, the $X$-ray transform of the characteristic function $\chi_K,$ i.e., the…

Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf…

In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C…

In this short note, we establish a quantitative description of the genericity of transversality of $C^1$-submanifolds in $\mathbb{R}^n$: Let $\Sigma \subset \mathbb{R}^n$ be a $d$-dimensional $C^1$-embedded submanifold where $n \geq d+1$.…

Suppose that to every non-degenerate simplex Delta in n-dimensional Euclidean space a `center' C(Delta) is assigned so that the following assumptions hold: (i) The map that assigns C(Delta) to Delta commutes with similarities and is…

The total diameter of a closed planar curve $C\subset R^2$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $C$. Furthermore, when $C$ is convex or centrally symmetric, the…

Let $K$ be a convex compact $GB$-subset of a separable Hilbert space $H$. Denote by $\mathrm{Spec}_k K$ the set $\{(\xi_1(h), \ldots, \xi_k(h))\colon h\in K\}\subset \mathbb{R}^k,$ where $\xi_1, \ldots, \xi_k$ are independent copies of the…

In this work we study the maximum relative diameter functional $d_M$ in the class of multi-rotationally symmetric planar convex bodies. A given set $C$ of this class is $k$-rotationally symmetric for $k\in\{k_1,\ldots,k_n\}\subset\nn$, and…

The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…

The c-functions, related to a reductive symmetric space G/H and a fixed representation of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a…

For a compact subset $K$ of the complex plane $\mathbb C,$ let $C(K)$ denote the algebra of continuous functions on $K$. For an open subset $U \subset K,$ let $A(K,U) \subset C(K)$ be the algebra of functions that are analytic in $U.$ We…

For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in…

Given a vector a $\in$ Rn, we provide an alternative and direct proof for the formula of the volume of sections delta $\cap$ {x : a T x \textless{}= t} and slices $\cap$ {x : a T x = t}, t $\in$ R, of the simplex delta. For slices the…

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds…

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups…

Two of the authors have defined the class $ WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and…