Related papers: A Simple Proof of Cauchy's Surface Area Formula

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more…

We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in…

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted…

The average section functional ${\rm as}(K)$ of a centered convex body in ${\mathbb R}^n$ is the average volume of central hyperplane sections of $K$: \begin{equation*}{\rm as}(K)=\int_{S^{n-1}}|K\cap \xi^{\perp }|\,d\sigma (\xi…

Let \Sigma be a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of \Sigma is bounded from below by the volume of the unit ball in R^k. This answers a question posed by R.…

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of…

We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain singularities, resolving these, and taking the double…

A convex surface contracting by a strictly monotone, homogeneous degree one function of curvature remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the…

We define the ``volume'' contained by pointed $k$-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed $k$-surface is always finite.

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity…

We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on…

Let X be a smooth cubic hypersurface. We prove that a general cubic surface is isomorphic to a hyperplane section of X .

Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed…

We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations $u_t=F(\n^2u, \n u, u, t)$ under a structural condition, and give a geometric lower bound of the principal…

We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence of a measure theoretic notion of…

We continue the investigation of formation of trapped surfaces in strongly curved , conformally flat geometries. Initial data in quasi-polar gauges rather then maximal ones are considered. This implies that apparent horizons coincide with…

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately…

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We find a necessary and sufficient condition that ensures a positive answer to…