The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

The Willmore functional and the containment problem in R4

Let Σ be a convex hypersurface in the Euclidean space R ⁴ with mean curvature H. We obtain a geometric lower bound for the Willmore functional ∫_Σ H ² dσ. This bound is an invariant involving the area of Σ, the volume and Minkowski quermassintegrals of the convex body that Σ bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R ⁴.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

The Number of Geometric Bistellar Neighbors of a Triangulation

The theory of secondary and fiber polytopes implies that regular (also called convex or coherent) triangulations of configurations with n points in R ^d have at least n-d-1 geometric bistellar neighbors. Here we prove that, in fact, all triangulations of n points in R ² have at least n-3 geometric bistellar neighbors. In a similar way, we show that for three-dimensional point configurations, in convex position and with no three points collinear, all triangulations have at least n-4 geometric bistellar flips. In contrast, we exhibit three-dimensional point configurations, with a single interior point, having deficiency on the number of geometric bistellar flips. A lifting technique allows us to obtain a triangulation of a simplicial convex 4-polytope with less than n-5 neighbors. We also construct a family of point configurations in R ³ with arbitrarily large flip deficiency. Received November 25, 1996, and in revised form March 10, 1997.     

Invariant measure for a class of parabolic degenerate equations

In this paper we consider a stochastic flow in Rⁿ which leaves a closed convex set K invariant. By using a recent characterization of the invariance, involving the distance function, we study the corresponding transition semigroup P_t and its infinitesimal generator N. Due to the invariance property, N is a degenerate elliptic operator. We study existence of an invariant measure ν of P_t and the realization of N in L² (H, ν).     

Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes

Let H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} denote the two dimensional hyperbolic space over \mathbb F{\mathbb F} , where \mathbb F{\mathbb F} is either the complex numbers \mathbb C{\mathbb C} or the quaternions \mathbb H{\mathbb H} . It is of interest to characterize algebraically the dynamical types of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} . For \mathbb F=\mathbb C{\mathbb F=\mathbb C} , such a characterization is known from the work of Giraud–Goldman. In this paper, we offer an algebraic characterization of isometries of H²_{\mathbb H}{{\bf H}^{\bf 2}_{\mathbb H}} . Our result restricts to the case \mathbb F=\mathbb C{\mathbb F=\mathbb C} and provides another characterization of the isometries of H²_{\mathbb C}{{\bf H}^{\bf 2}_{\mathbb C}} , which is different from the characterization due to Giraud–Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of H²_{\mathbb F}{{\bf H}^{\bf 2}_{\mathbb F}} and determine the z-classes.     

Compact,convex sets in R n and a new banach lattice. I.- Theory.

The present paper shows that compact, non-empty convex sets in R ⁿ form a wedge in a well-defined Banach lattice, which turns out to be isometrically Riesz-isomorphic to the continuous functions in S ^n–1, the unit sphere of R ⁿ . Among other results, we obtain Dini-like convergence results for sets, linking order- and norm-convergence.     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

Uniqueness of horospheres and geodesic cylinders in hyperbolic space

A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ? Hⁿ⁺¹ be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further H_r ≡ c for one normalized mean curvature. Then Mⁿ is a horosphere or a geodesic cylinder if $r{\leq}\ {2\over 3}\ (n+1)$ . For $r >\ {2\over 3}\ (n+1)$ the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions.     

On the Volume Formulas of Cones and Orthogonal Multi-cones in S^n（1） and H^n（-1）

Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sⁿ(1) and Hⁿ(—1). (Dedicated to the memory of Shiing-Shen Chern)     

A note on the sufficient initial condition ensuring the convergence of directly extended 3-block ADMM for special semidefinite programming

ABSTRACT

In this note, we consider three types of problems, H-weighted nearest correlation matrix problem and two types of important doubly non-negative semidefinite programming, derived from the binary integer quadratic programming and maximum cut problem. The dual of these three types of problems is a 3-block separable convex optimization problem with a coupling linear equation constraint. It is known that, the directly extended 3-block alternating direction method of multipliers (ADMM3d) is more efficient than many of its variants for solving these convex optimization, but its convergence is not guaranteed. By choosing initial points properly, we obtain the convergence of ADMM3d for solving the dual of these three types of problems. Furthermore, we simplify the iterative scheme of ADMM3d and show the equivalence of ADMM3d to the 2-block semi-proximal ADMM for solving the dual's reformulation, under these initial conditions.     

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 .     

Characterization and recognition of d.c. functions

A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C ^1,1 functions and lower-C ² functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.     

Eine Charakterisierung der Sphäre im E3

It is well known, that in E³ the spheres are the closed convex C² -surfaces having the property, that each of their closed C²-curves with the total geodesic curvature O bisects the area of the surface. This characterization will be transmitted into the theory of convex surfaces founded by A.D.Alexandrov, where convex surfaces without any differentiability property are studied.     

Approximating Minimum-Weight Triangulations in Three Dimensions

Let S be a set of noncrossing triangular obstacles in R ³ with convex hull H . A triangulation T of H is compatible with S if every triangle of S is the union of a subset of the faces of T. The weight of T is the sum of the areas of the triangles of T. We give a polynomial-time algorithm that computes a triangulation compatible with S whose weight is at most a constant times the weight of any compatible triangulation. One motivation for studying minimum-weight triangulations is a connection with ray shooting. A particularly simple way to answer a ray-shooting query (``Report the first obstacle hit by a query ray') is to walk through a triangulation along the ray, stopping at the first obstacle. Under a reasonably natural distribution of query rays, the average cost of a ray-shooting query is proportional to triangulation weight. A similar connection exists for line-stabbing queries (``Report all obstacles hit by a query line'). Received February 3, 1997, and in revised form August 21, 1998.     

A Brunn-Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in R³, as an application we get the Brunn-Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in R³.