Smooth convex limit surfaces in the space of symmetric second-rank tensors

Equations are proposed to describe smooth (regular) convex hexagonal and triangular curves in the deviatoric plane. The equations are used to construct smooth convex limit surfaces for media whose properties are arbitrarily anisotropic. The results supplement the findings in [A. Lagzdin' and A. Zilauts, Mekh. Kompozitn. Mater.,32, No. 3, 339–349 (1996)], where the third invariant of a tensor was introduced into the limit-surface equation by means of singular deviatoric curves.Institute of Polymer Mechanics of the Latvian Academy of Sciences, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 171–181, March–April, 1997.     

Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

Given two densities on with the same total mass, the Monge transport problem is to find a Borel map rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.

     

Bending of convex surfaces

A survey of results on bending of nonregular convex surfaces is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 10, pp. 193–224, 1978.     

Capillary surfaces in a convex cone

We show that a compact embedded hypersurface ${S\subset\mathbb{R}^{n+1}}$ with constant higher-order mean curvature in a convex piecewise smooth cone C which is perpendicular to ?C is part of a hypersphere. Also we prove that an embedded disk type constant mean curvature surface ${S\subset\mathbb{R}^3}$ in a convex polyhedral cone C which makes constant contact angles with ?C is a spherical cap if C has at most five faces. This condition on the number of faces can be dropped if C is a right cone over a regular n-gon and the contact angles are the same on ?S.     

On typical degenerate convex surfaces

Various properties are given concerning geodesics on, and distance functions from points in, typical degenerate convex surfaces; i.e., surfaces obtained by gluing together two isometric copies of typical (in the sense of Baire category) convex bodies, by identifying the corresponding points of their boundaries.     

Uniqueness theorem for convex surfaces

A proof is given of the following assertion: two closed convex analytic surfaces in three-dimensional Euclidean space are equal if their areas and lengths of boundaries of orthogonal projections onto any plane coincide.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 115–117, July, 1969.     

The central limit problem for convex bodies

It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.

     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that

The limit shape of convex lattice polygons

It is proved here that, asn→∞, almost all convex (1/n)ℤ²-lattice polygons lying in the square [−1, 1]² are very close to a fixed convex set. This research was partially supported by Hungarian Science Foundation Grants 1907 and 1909.     

Infinite curvature on typical convex surfaces

Solving a long-standing open question of Zamfirescu, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of Alexandrov spaces of bounded curvature, and show continuity properties for this notion.     

Remote limit points on surfaces

A flow (continuous real action) on a compact orientable surface M of genus greater than one (a sphere with at least two handles) has sufficient room for orbits to wrap around one of the handles in an exotic fashion. Specifically, an orbit that is wrapping around one handle can, between wraps, spend increasing amounts of time wrapping and unwrapping around a second handle before returning to the first for the next wrap around it. As a result the omega limit set of such an orbit can contain a simple closed curve of fixed points around the second handle in spite of wrapping around the first handle. In an earlier paper (Colloq. Math. 84/85 (2000) 235), the authors constructed such a flow from this perspective and studied its lift to the universal covering space of the surface. In this paper it is shown that many of the properties of the example are consequences of a general theory that extends classical limit cycle theory.