Related papers: On convex bodies with constant non-central section…

Generalizing the slicing inequality for functions on convex bodies from [11], it was proved in [4] that there exists an absolute constant $c$ so that for any $n\in \mathbb N$, any $q\in [0,n-1)$ which is not an odd integer, any…

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function…

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…

The following result is proved: {\bf Theorem.} Let $D\subset \R^3$ be a bounded domain homeomorphic to a ball, $|D|$ be its volume, $|S|$ be the surface area of its smooth boundary $S$, $D\subset B_R:=\{x:|x|\leq R\}$, and $H_R$ is the set…

It is natural to ask for a reasonable constant k having the property that any open set of area greater than k on a bidimensional sphere of area 1 always contains the vertices of a regular tetrahedron. We shall prove that it is sufficient to…

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…

Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface…

We show that there exist convex bodies of constant width in $\mathbb{E}^n$ with illumination number at least $(\cos(\pi/14)+o(1))^{-n}$, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter $1$ in…

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…

In this paper, we study the quaternionic counterpart of complex Fock spaces $\mathfrak{F}_{\alpha}^p ( 0<p<\infty$ and for some parameter $\alpha$) of entire slice hyperholomorphic functions in an Euclidean unit ball $\mathbb{B}^n$ in…

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$ in the Euclidean plane, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$…

Let $M$ be a 4-dimensional open manifold with nonnegative Ricci curvature. In this paper, we prove that if the universal cover of $M$ has Euclidean volume growth, then the fundamental group $\pi_1(M)$ is finitely generated. This result…

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms…

Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…

It was shown by G. Pisier that any finite-dimensional normed space admits an $\alpha$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball…

In this paper we consider the equiform motion of a helix in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant.…

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive…

Complete hyperbolicity of small Euclidean balls with respect to a C^1-smooth almost complex structure standard at origin is improved to give a complete hyperbolicity of strictly pseudoconvex domains. More precise (and lower) regularity…

We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and…

In this paper we present a unique 4-dimensional body of constant width based on the classical notion of focal conics.