相关论文: A reverse isoperimetric inequality for planar ($\a…

We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded $\lambda$-convex curve of a given length on the Lobachevsky plane.

Tube-like surfaces are widely encountered in geometry processing, engineering structures, and medical anatomy, yet their intrinsic longitudinal and circumferential topology is not well preserved by conventional planar annular or rectangular…

We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal\'o point is at the origin, with those of its inner and outer regularizations. We also provide functional…

The aim of this paper is to present some properties of reduced spherical convex bodies on the two-dimensional sphere $S^2$. The intersection of two different non-opposite hemispheres is called a lune. By its thickness we mean the distance…

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…

We study conformal quantum mechanics by first considering the perturbative $S$-matrix in various dimensions. The model has two couplings and we study perturbatively the degree of ultraviolet divergences arising in the interplay between the…

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there…

We study the $O_\beta$-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle $\beta$. Given a set $P$ of $n$ points in the plane, we show how to maintain the $O_\beta$-hull of $P$…

The well-studied vector balancing constant $\beta(U, V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $\alpha(U,V)$. In [BS97], Banaszczyk and Szarek proved that $\alpha(B_2^n, V)\leq c$ when $V$ has…

For a domain $\Omega$ in a geodesically convex surface, we introduce a scattering energy $\mathcal{E}(\Omega)$, which measures the asymmetry of $\Omega$ by quantifying its incompatibility with an isometric circle action. We prove several…

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies obtained for linear systems without zero order term in bounded domains and…

We show the uniqueness of the radial centers of any order $\alpha$ of a parallel body of a convex body $\Omega$ in $\mathbb R^m$ at distance $\delta$ if $\delta$ is greater than the diameter of $\Omega$ multiplied by a constant which…

We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities…

In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented…

We give a complete classification of homogeneous $(\alpha,\beta)$-metrics with positive flag curvature and vanishing S-curvature.

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces…

The famous Minkowski inequality provides a sharp lower bound for the mixed volume $V(K,M[n-1])$ of two convex bodies $K,M\subset\mathbb{R}^n$ in terms of powers of the volumes of the individual bodies $K$ and $M$. The special case where $K$…

We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the…

Consider a solution $f\in C^{2}(\Omega)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ \Omega\subset R^{2}, \] where $\Omega$ is a domain whose…

In this paper, an inverse initial-boundary value problem for the heat equation in three dimensions is studied. Assume that a three-dimensional heat conductive body contains several cavities of strictly convex. In the outside boundary of…