Starshaped hypersurfaces¶and the mean curvature flow

Under the assumption of two a-priori bounds for the mean curvature, we are able to generalize a recent result due to Huisken and Sinestrari [8], valid for mean convex surfaces, to a much larger class. In particular we will demonstrate that these a-priori bounds are satisfied for a class of surfaces including meanconvex as well as starshaped surfaces and a variety of manifolds that are close to them. This gives a classification of the possible singularities for these surfaces in the case n= 2. In addition we prove that under certain initial conditions some of them become mean convex before the first singularity occurs. Received: 6 June 1997 / Revised version: 24 October 1997     

A NOTE ON THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.     

Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations

We derive bounds on the expectation of a class of periodic functions using the total variations of higher-order derivatives of the underlying probability density function. These bounds are a strict improvement over those of Romeijnders et al. (Math Program 157:3–46, 2016b), and we use them to derive error bounds for convex approximations of simple integer recourse models. In fact, we obtain a hierarchy of error bounds that become tighter if the total variations of additional higher-order derivatives are taken into account. Moreover, each error bound decreases if these total variations become smaller. The improved bounds may be used to derive tighter error bounds for convex approximations of more general recourse models involving integer decision variables.     

A general construction of barycentric coordinates over convex polygons

Barycentric coordinates are unique for triangles, but there are many possible generalizations to convex polygons. In this paper we derive sharp upper and lower bounds on all barycentric coordinates over convex polygons and use them to show that all such coordinates have the same continuous extension to the boundary. We then present a general approach for constructing such coordinates and use it to show that the Wachspress, mean value, and discrete harmonic coordinates all belong to a unifying one-parameter family of smooth three-point coordinates. We show that the only members of this family that are positive, and therefore barycentric, are the Wachspress and mean value ones. However, our general approach allows us to construct several sets of smooth five-point coordinates, which are positive and therefore barycentric. Dedicated to Charles A. Micchelli on his 60th Birthday Mathematics subject classifications (2000) 26C15, 65D05.     

Applications of a semi-dynamic convex hull algorithm

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.     

On the role of convexity in isoperimetry,spectral gap and concentration

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-émery. Supported by NSF under agreement #DMS-0635607.     

On the mean number of normals through a point in the interior of a convex body

Recently, Kathy Hann established bounds on the average number of normals through a point in a convex bodyK, in the cases whereK is either a polytope or sufficiently smooth. In addition, an Euler-type theorem was obtained for these particular classes of convex bodies. In the present work we show that all these statements are true for an arbitrary convex bodyK. For this purpose measure geometric tools and a general approximation technique will be essential.     

Fair upper bounds for the curvature in univariate convex interpolation

In convex interpolation the curvature of the interpolants should be as small as possible. We attack this problem by treating interpolation subject to bounds on the curvature. In view of the concexity the lower bound is equal to zero while the upper bound is assumed to be piecewise constant. The upper bounds are called fair with respect to a function class if the interpolation problem becomes solvable for all data sets in strictly convex position. We derive fair a priori bounds for classes of quadraticC ¹, cubicC ², and quarticC ³ splines on refined grids.     

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ³ which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ⁿ with nonvanishing Gaussian curvature.Written with partial supports from NSF grant No. MCS-8211461.     

A-priori error estimates of Galerkin backward differentiation methods in time-inhomogeneous parabolic problems

Summary Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.     

Linear and convex aggregation of density estimators

We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.      

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x ^* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x^*)-LB)/{f(x^*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.     

The geometry of logconcave functions and sampling algorithms

The class of logconcave functions in ℝⁿ is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from logconcave density functions, we study their geometry and introduce a technique for “smoothing” them out. These results are applied to analyze two efficient algorithms for sampling from a logconcave distribution in n dimensions, with no assumptions on the local smoothness of the density function. Both algorithms, the ball walk and the hit‐and‐run walk, use a random walk (Markov chain) to generate a random point. After appropriate preprocessing, they produce a point from approximately the right distribution in time O*(n⁴) and in amortized time O*(n³) if n or more sample points are needed (where the asterisk indicates that dependence on the error parameter and factors of log n are not shown). These bounds match previous bounds for the special case of sampling from the uniform distribution over a convex body.© 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Convolutions for special classes of harmonic univalent functions

Ruscheweyh and Sheil-Small proved the PólyarSchoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that close-to-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve close-to-convexity under convolution. An auxiliary function enables us to obtain necessary and sufficient convolution conditions for convex and starlike harmonic functions, which lead to sufficient coefficient bounds for inclusion in these classes.     

Erdős–Szekeres Theorem for Lines

According to the Erd?s–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd?s–Szekeres theorem.     

Convex hulls of curves of genus one

Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials fi with (mod IC) and such that the degrees deg(fi) are bounded in terms of deg(f) only. Using Lasserre?s relaxation method, we deduce an explicit representation of the convex hull of C(R) in Rn by a lifted linear matrix inequality. This is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety. The same works for convex hulls of (singular) curves whose normalization is C. We then make a detailed study of the associated degree bounds. These bounds are directly related to size and dimension of the projected matrix pencils. In particular, we prove that these bounds tend to infinity when the curve C degenerates suitably into a singular curve, and we provide explicit lower bounds as well.     

On ordinary points in arrangements

We show that in an arrangement ofn curves in the plane (or on the sphere) there are at leastn/2 points where precisely 2 curves cross (ordinary points). Furthermore there are at least (4/3)n triangular regions in the complex determined by the arrangement. Triangular regions and ordinary vertices are both connected with boundary vertices of certain distinguished subcomplexes. By analogy with rectilinear planar polygons we distinguish concave and convex vertices of these subcomplexes. Our lower bounds arise from lower bounds for convex vertices in the distinguished subcomplexes.     

The effect of transformations on the approximation of univariate (convex) functions with applications to Pareto curves

In the literature, methods for the construction of piecewise linear upper and lower bounds for the approximation of univariate convex functions have been proposed. We study the effect of the use of transformations on the approximation of univariate (convex) functions. In this paper, we show that these transformations can be used to construct upper and lower bounds for nonconvex functions. Moreover, we show that by using such transformations of the input variable or the output variable, we obtain tighter upper and lower bounds for the approximation of convex functions than without these approximations. We show that these transformations can be applied to the approximation of a (convex) Pareto curve that is associated with a (convex) bi-objective optimization problem.