相关论文: Strengthened Euler's Inequality in Spherical and H…

We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase.…

A recent paper [CGT] studies the evolution of star-shaped mean convex hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding curvature flows. In the present paper we consider the same problem in the real, complex and…

Most research into similarity search in metric spaces relies upon the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common…

In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the…

After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…

Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of…

A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…

While geometry with transcendental curves, like the Quadratrix of Hippias and the Spiral of Archimedes, played a significant role in our modern developments of geometry and algebra. The investigation has fallen off in the modern era despite…

In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear…

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the…

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side;…

In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a…

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is…

Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ have been…

The reduced system in the problem of the inertial motion of a rigid body with a fixed point (the Euler case) is equivalent, by the Maupertuis principle, to some geodesic flow on the 2-sphere. We describe the phase topology of this case…

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the…

We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections…

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-\emph{hyperbolic} $($in the…

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to…