Related papers: The equal tangents property

Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…

Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are…

We prove $\epsilon$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $\delta$-small compared to the mean curvature. We give the explicit dependence of $\delta$ on…

Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$…

Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.

For a specific class of surfaces of revolution S, the existence of a smooth map {\Phi} from a neighbourhood U of S to the Euclidean plane E2 preserving distances infinitesimally along the meridians and the parallels of S and sending the…

Let $P$ and $Q$ be two convex polytopes both contained in the interior of an Euclidean ball $r\textbf{B}^{d}$. We prove that $P=Q$ provided that their sight cones from any point on the sphere $rS^{d-1}$ are congruent. We also prove an…

Let us have in S^2, R^2 or H^2 a pair of convex bodies, for S^2 different from S^2, such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any…

This paper is devoted to studying of some properties of multivalued mappings in Euclidean space. There were proved theorems on a fixed point for multivalued mappings whose restrictions to some subset in the closure of a domain satisfy "a…

We give another proof of a theorem of Scharlemann and Tomova and of a theorem of Hartshorn. The two theorems together say the following. Let M be a compact orientable irreducible 3--manifold and P a Heegaard surface of M. Suppose Q is…

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on…

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped…

We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the…

One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…

A point P on a smooth hypersurface X of degree d in an N-dimensional projective space is called a star point if and only if the intersection of X with the embedded tangent space T_P(X) is a cone with vertex P. This notion is a…

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the…

This purpose of this paper is to prove the following result: let phi be a strictly convex, smooth, convex body in the Euclidean plane, if the intersection of n translates of phi has a non-empty interior, and all of the translates contribute…

We do further investigation in a certain cosine function defined for smooth Minkowski spaces. We prove that such function is symmetric if and only if the referred space is Euclidean, and also that it can be given in terms of the Gateaux…

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of…