Smooth Approximation of Lipschitz Maps and Their Subgradients

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterisation of the basic open subsets of the L-topology in terms of ties or primitive maps of functions. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative.

Isometric surfaces share the same geometric structure also known as the first fundamental form. For example, bending of a given surface, that includes length preserving deformations without tearing or stretching the surface, are considered to be isometric. We present a method to construct a bending invariant canonical form for such surfaces. This invariant representation is an embedding of the intrinsic geodesic structure of the surface in a finite dimensional Euclidean space, in which geodesic distances are approximated by Euclidean ones. The canonical representation is constructed by first measuring the intergeodesic distances between points on the surfaces. Next, multi-dimensional scaling (MDS) techniques are applied to extract a finite dimensional flat space in which geodesic distances are represented as Euclidean ones. The geodesic distances are measured by the efficient fast marching on triangulated domains numerical algorithm. Applying this transform to various objects with similar geodesic structures (similar first fundamental form) maps isometric objects into similar canonical forms. We show a simple surface classification method based on the bending invariant canonical form.

Representations of cyclic groups acting on complete simplicial fans

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Background. The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds. Objective. The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae. Methods. Basic results of mathematical analyses, measure theory and differential geometry are used. Results. An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space is obtained. It is shown that such approach is consistent with the classical definition of the surface area. Conclusions. The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

Separating decompositions of metric spaces are an important randomized clustering paradigm that was formulated by Bartal in [Bar96] and is defined as follows. Given a metric space (X, dx), its modulus of separated decomposability, denoted SEP(X, dx), is the infimum over those σ ∊ (0,∞] such that for every finite subset S ⊆ X and every Δ > 0 there exists a distribution over random partitions P of S into sets of diameter at most Δ such that for every x,y ∊ S the probability that both x and y do not fall into the same cluster of the random partition P is at most σdx (x, y)/Δ. Here we obtain new bounds on SEP(X, ‖·‖x) when (X, ‖· ‖x) is a finite dimensional normed space, yielding, as a special case, that for every n ∊ N. More generally, for every p ∊ [2, ∞]. This improves over the work [CCG+ 98] of Charikar, Chekuri, Goel, Guha, and Plotkin, who obtained this bound when p = 2, yet for p ∊ (2, ∞] they obtained the asymptotically weaker estimate One should note that it was claimed in [CCG+98] that the bound is sharp for every p ∊ [2, ∞], and in particular it was claimed in [CCG+98] that However, the above results show that this claim of [CCG+98] is incorrect for every p ∊ (2, ∞]. Our new bounds on the modulus of separated decomposability rely on extremal results for orthogonal hyperplane projections of convex bodies, specifically using the work [BN02] of Barthe and the author. This yields additional refined estimates, an example of which is that for every n ∊ ℕ and k ∊ {1,…,n} we have where denotesthe subset of ℝn consisting of all those vectors that have at most k nonzero entries, equipped with the Euclidean metric.The above statements have implications to the Lip- schitz extension problem through its connection to random partitions that was developed by Lee and the author in [LN04, LN05]. Given a metric space (X,dx), let e(X) denote the infimum over those K ∊ (0, ∞] such that for every Banach space Y and every subset S ⊆ X, every 1-Lipschitz function f : S → Y has a K-Lipschitz extension to all of X. Johnson, Lindenstrauss and Schechtman proved in [JLS86] that e(X) < dim(X) for every finite dimensional normed space (X, ‖ · ‖x). It is a longstanding open problem to determine the correct asymptotic dependence on dim(X) in this context, with the best known lower bound, due to Johnson and Lindenstrauss [JL84], being that the quantity e(X) must sometimes be at least a constant multiple of In particular, the previously best known upper bound on e(ℓn∞) was the O(n) estimate of [JLS86]. It is shown here that for every n ∊ ℕ we have thus answering (up to logarithmic factors) a question that was posed by Brudnyi and Brudnyi in [BB05, Problem 2]. More generally, for every p ∊ [2, ∞], thus resolving (negatively) a conjecture of Brudnyi and Brudnyi in [BB05, Conjecture 5].

The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to find the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in finite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

Introduction. This paper gives counterexamples for two unrelated conjectures which pertain to convexity theory. Supposef is any function defined on a closed convex set C in finitedimensional Euclidean space. Its envelope function env f is defined as the pointwise supremum of all linear functions which f dominates everywhere. It is elementary and well known that env f is convex, and hence continuous in the relative interior of C. It is similarly known that env f is lower-semicontinuous on the boundary of C. Witsenhausen [14] uses and studies envelope functions. He suggested the following orally: CONJECTURE 1. Iff is continuous everywhere on C, then envf is also continuous everywhere on C. I provide a very simple counterexample in 3-space which shows that this is not true. Incidentally, though the proof is not on paper, I feel fairly certain that Witsenhausen's conjecture is correct in 2-space. If C is a closed convex set in finite-dimensional Euclidean space, then for any point x in the space there is a unique point P(x) in C which is nearest to x. The mapping P is called the projection onto C. (Many interesting results about this mapping are given in Phelps [9], [10].) It is easy to prove and well known that P is a contraction mapping, hence continuous. It is also easy to see that P need not have much in the way of differentiability properties, for if C is a polyhedron in 2-space, then it is easy to find points and directions at which P does not even have a two-sided directional derivative. The existence in general of a one-sided directional derivative, however, is not so easy to decide. This question does not seem to be raised in such natural places as Eggleston [4] and Valentine [11]. CONJECTURE 2. For all points and directions, P has a one-sided directional derivative. My mildly complicated counterexample is a set C (in 3-space), which is the convex hull of a countable infinity of points. I suspect that the conjecture is correct in 2-space. In n-space it would be interesting to know whether the set of points and directions at which the conjecture holds is dense in EnXSn-. If not, the convex set would have to be extremely jagged!

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k &amp;amp;gt; 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

We study the Pareto frontier for two competing norms $\|\cdot\|_X$ \nomenclature[X]$\lVert\cdot\rVert_X,\lVert\cdot\rVert_Y$norms and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the Pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The singular value decomposition of a matrix is closely related to the Pareto frontier for the spectral and nuclear norm. We will develop a general theory that extends the notion of singular values of a matrix to arbitrary finite dimensional Euclidean vector spaces equipped with dual norms. This also generalizes the diagonal singular value decompositions (DSVDs) for tensors introduced by the author in previous work. We can apply the results to denoising, where $c$ is a noisy signal, $a$ is a sparse signal, and $b$ is noise. Applications include 1D total variation denoising, 2D total variation Rudin--Osher--Fatemi image denoising, LASSO, basis pursuit denoising, and tensor decompositions.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a sharp edge on the unit sphere in B*. The attempt to describe a sharp edge of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer’s theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer’s theorem is necessary in the theory of stochastic nonlinear operators.

The systematic study of various mean-values of nonnegative valued functions is due in large part to the work of J. L. W. V. Jensen [12], A. Kolmogoroff [11], and B. Jessen [13]. In this paper we present a similar theory of certain means of functions whose values are taken from a collection of convex bodies in a finite dimensional Euclidean space. The range of nonnegative valued functions is totally ordered while the collection of convex bodies to be considered is partially ordered by set inclusion; corresponding to inequalities, we shall have inclusions, here to be called inequalities, between mean-values. Thus the discussion will furnish an example of a partially ordered system whose algebraic and topological structure is sufficient to admit a fairly elaborate theory of inequalities analogous to that of the real numbers. The first section is a resume of those parts of the work of Jensen, Kolmogoroff, and Jessen for which analogues will be developed. ?2 treats of pertinent material about star and convex bodies. Certain families of star bodies and of convex bodies are defined in the third section; these latter play the role of nonnegative valued functions. Two systems of power means of such families are described. Each system is, in its way, anialogous to the power means of nonnegative valued functions. In ?4 we discuss some special cases of power means including elementary means and the Riemaiin-Minkowski integrals of A. Dinghas [2], [3]. Also some rotation invariants of a convex body are described as power means of special families determined by the convex body. The fifth section begins with crucial properties of the means and contains an extremal characterization for the two systems of means defined in the third section. Analogues of Jensen's and Jessen's inequalities make up ?6. We also mentioni the limiting cases of power means as the power index becomes infinite, positively and negatively. As an applicationl of Jessen's inequality, a Brunn-Minkowski type theorem is proved. In the final section we discuss some further systems of meanls of convex bodies. 1. The definition of the elementary power means extends in a natural way to include power means of certain functions f which have finite, nonnegative values and whose domain T is a compact topological group equipped with its Haar measure. Iff is bounded and measurable over its domain, its pth- power mean is defined by

Finite groups admitting grassmannian 4-designs

In 1965, Shepard introduced a general interpolation formula for arbitrarily spaced data over any finite dimensional Euclidean space. The deficiencies in his initial results have been corrected and later derivations have been widely applied to surface fitting problems in aviation design and geology.This paper describes a new research direction in Shepard's method and a new application to computer graphics. In the first, the convex weighting functions are generalized by the creation of a user-defined mathematical domain which is independent of the surface. In the second, the original formula is applied to blend mapping projections for aviation visual simulation and an algorithm for rapid evaluation is presented.