Shortest paths on convex hypersurfaces of Riemannian spaces

A convex hypersurface in a Riemannian space M^m is part of the boundary of an m-dimensional locally convex set. It is established that there exists an intrinsic metric of such a hypersurface and it has curvature which is bounded below in the sense of A. D. Aleksandrov; curves with bounded variation of rotation in are shortest paths in M^m. For surfaces in R^m these facts are well known; however, the constructions leading to them are in large part inapplicable to spaces M^m. Hence approximations to by smooth equidistant (not necessarily convex) ones and normal polygonal paths, introduced (in the case of R³) by Yu. F. Borisov are used.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 114–132, 1976.     

Rectangular convexity of convex domains of constant width

An E R ² is r-convex if for every x, y E there exists a closed rectangle R such that x, y R and R E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

On square integrability of mean curvature for surfaces with positive Gaussian curvature

We show that {ie319-1} H ²dµ = for any complete surface M R ³ which has positive curvature outside a compact subset of R ³. This proves a conjecture of Friedrich.     

Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of theintersections of K with all its possible translates. Matheronconjectured in 1986 that this information determines K amongall convex bodies, up to certain known ambiguities. It is provedthat this is the case if K R² is not C¹, or it is not strictlyconvex, or its boundary contains two arbitrarily small C² openportions ‘on opposite sides’. Examples are alsoconstructed that show that this conjecture is false in Rⁿ forany n 4.     

Hilbert Geometry for Strictly Convex Domains

We prove in this paper that the Hilbert geometry associated with a bounded open convex domain in R ⁿ whose boundary is a ² hypersuface with nonvanishing Gaussian curvature is bi-Lipschitz equivalent to the n-dimensional hyperbolic space H ⁿ . Moreover, we show that the balls in such a Hilbert geometry have the same volume growth entropy as those in H ⁿ .     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

Convexes hyperboliques et fonctions quasisymétriques

Hyperbolic convex sets and quasisymmetric functions Every bounded convex open set of R ^m is endowed with its Hilbert metric d . We give a necessary and sufficient condition, called quasisymmetric convexity, for this metric space to be hyperbolic. As a corollary, when the boundary is real analytic, is always hyperbolic.In dimension 2, this condition is: in affine coordinates, the boundary is locally the graph of a C¹ strictly convex function whose derivative is quasisymmetric.      

Isometric Immersions of Closed Manifolds of Nonnegative Curvature

Let M be a closed manifold and let be an immersion inducing a C²-smooth (respectively, polyhedral) metric of nonnegative curvature on M. If this nonnegativity property is preserved under all affine transformations of , then f is an embedding into the boundary of a C²-smooth convex body (respectively, a convex polyhedron) in a certain . Bibliography: 6 titles.     

Anisotropic curvature-driven flow of convex sets

We study in this paper the mean curvature evolution, and in particular the anisotropic mean curvature evolution, of convex sets in R^N${\mathbb{R}}^N$ (without driving forces). If the anisotropy is smooth, we show that the evolution remains convex. If the anisotropy is crystalline, we build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

A counterexample to a ‘simplex algorithm’ for convex bodies

A counterexample, in E ³, is given to the following conjecture. Suppose f ^* is a linear functional, and e an exposed point of a convex body K such that f ^* does not attain its maximum on K at e; then there is an f ^*-strictly increasing path in the one-skeleton of K emanating from e. The counterexample shows that a certain generalized simplex algorithm fails. Furthermore for a different linear functional f, there are no three disjoint f-strictly increasing paths in the one-skeleton of K leading to e.     

Über die Approximation von lokal konvexen Mengen

While convex sets in Euclidean space can easily be approximated by convex sets with C -boundary, the C -approximation of convex sets in Riemannian manifolds is a non-trivial problem. Here we prove that C-approximation is possible for a compact, locally convex set C in a Riemannian manifold if (i) C has strictly convex boundary or if (ii) the sectional curvature is positive or negative on C.The proofs are based on a detailed analysis of the distance function from C, on results from [1] and on the Greene-Wu approximation process for convex functions ([5], [6]). Finally, using similar methods, a partial tubular neighborhood with geodesic fibres is constructed for the boundary of a locally convex set. This construction is essential for some results in [2].     

Approximation by Homogeneous Polynomials

Let be the boundary of a convex domain symmetric to the origin. The conjecture that any continuous even function can be uniformly approximated by homogeneous polynomials of even degree on K is proven in the following cases: (a) if d = 2; (b) if K is twice continuously differentiable and has positive curvature in every point; or (c) if K is the boundary of a convex polytope.     

Singularity of mean curvature flow of Lagrangian submanifolds

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold ₀ is Lagrangian and almost calibrated by Re in a Calabi-Yau n-fold (M,), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X₀,T) is a stationary Lagrangian integer multiplicity current in R²ⁿ with volume density greater than one at X₀. When n=2, the tangent cone is a finite union of at least two 2-planes in R⁴ which are complex in a complex structure on R⁴.     

Radial graphs of constant curvature and prescribed boundary

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, without assuming the convexity of the prescribed solution and using the theory of fully nonlinear elliptic equations. If the given data admits a suitable radial graph as a subsolution, then we prove that there exists a radial graph with constant curvature and realizing the prescribed boundary. As an application, it is proved that if \(\Omega \subset \mathbb {S}^n\) is a mean convex domain whose closure is contained in an open hemisphere of \(\mathbb {S}^n\) then, for \(0<R<n(n-1),\) there exists a radial graph of constant scalar curvature R and boundary \(\partial \Omega \).     

The Randomized Integer Convex Hull

Let K R^d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I_L(K) = conv (K L), where L is a randomly translated and rotated copy of the integer lattice Z^d. We estimate the expected number of vertices of I_L(K), whose behaviour is similar to the expected number of vertices of the convex hull of Vol K random points in K. In the planar case we also describe the expectation of the missed area Vol (K \ I_L(K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.     

A new characterization of convex plates of constant width

A convex plate DR ² of diameter 1 is of constant width 1 if and only if any two perpendicular intersecting chords have total length 1.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 1238.     

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Let be the tiling of R ³ with unit cubes whose vertices belong to the fundamental lattice L ₁ of points with integer coordinates. Denote by L _n the lattice consisting of all points x in R ³ such that nx belongs to L ₁. When the vertices of a polyhedron P in R ³ are restricted to lie in L ₁ then there is a formula which relates the volume of P to the numbers of all points of two lattices L ₁ and L _n lying in the interior and on the boundary of P. In the simplest case of the lattices L ₁ and L ₂ there are 27 points in each cube from whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R ³ which involves only nine points in each cube of : one from L ₂ and eight belonging to L ₄. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.