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

In this paper we show that a spherical convex body $C$ is of constant diameter $\tau$ if and only if $C$ is of constant width $\tau$, for $0<\tau<\pi$. Moreover, some applications to Wulff shapes are given.

Consider a uniform variate on the unit upper-half sphere of dimension $d$. It is known that the straight-line projection through the center of the unit sphere onto the plane above it distributes this variate according to a $d$-dimensional…

In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean…

In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a…

We compute the area of a generic d-sphere in a Snyder geometry.

In the present paper we establish area and volume estimates for spacetimes satisfying the strong energy condition in terms of the area and the $L^n$-norm of the second fundamental form or the mean curvature of an initial Cauchy…

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded…

For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the…

For asymptotically flat spacetimes, using the inverse mean curvature flow, we show that any compact $2$-surface, $S_0$, whose mean curvature and its derivative for outward direction are positive in spacelike hypersurface with non-negative…

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $\Sigma$ is a complete, properly…

In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $\alpha$-power of positive part of Gauss Curvature. First, for 1/2<\alpha\leq 1$, we show that…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

It is proved that if $u_1,\ldots, u_n$ are vectors in ${\Bbb R}^k, k\le n, 1 \le p < \infty$ and $$r = ({1\over k} \sum ^n_1 |u_i|^p)^{1\over p}$$ then the volume of the symmetric convex body whose boundary functionals are $\pm u_1,\ldots,…

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a…

We define an attractive gravity probe surface (AGPS) as a compact 2-surface $S_\alpha$ with positive mean curvature $k$ satisfying $r^a D_a k / k^2 \ge \alpha$ (for a constant $\alpha>-1/2$) in the local inverse mean curvature flow, where…

In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly…

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the…

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after…

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and…

This paper aims to develop the mathematical representation of a surface generated by elliptical arcs joining the sides of a regular polygon to a point lying vertically upward on the central axis of the polygon. The volume of the…