scholarly journals An Integralgeometric Approach to Dorronsoro Estimates

2019 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Function ◽  
Parabolic Problem ◽  
Elementary Proof ◽  
Lipschitz Functions ◽  
Geometric Proof ◽  
Proof Technique ◽  
Almost Everywhere ◽  
Small Scales ◽  
Affine Functions ◽  
Similar Proof

Abstract A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro’s theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher’s theorem for parabolic Lipschitz functions.

Fr ´Echet Differentiability Except For Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Spaces ◽  
Common Point ◽  
Lipschitz Functions ◽  
Infinite Dimensional ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Gateaux Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

scholarly journals Lipschitz functions with unexpectedly large sets of nondifferentiability points

2005 ◽  
Vol 2005 (4) ◽  
pp. 361-373 ◽  
Author(s):  
Marianna Csörnyei ◽  
David Preiss ◽  
Jaroslav Tišer
Keyword(s):  
Lipschitz Function ◽  
Measure Zero ◽  
Common Point ◽  
Lipschitz Functions ◽  
Large Sets ◽  
Set Of Points ◽  
Dense Set

It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.

scholarly journals Approximation of functions by means of Lipschitz functions

1963 ◽  
Vol 3 (2) ◽  
pp. 134-150 ◽  
Author(s):  
J. H. Michael
Keyword(s):  
Lipschitz Function ◽  
Unit Cube ◽  
Lipschitz Functions ◽  
Approximation Of Functions ◽  
Image Position ◽  
Elementary Area

Let Q denote the closed unit cube in Rn. The elementary area A(f) of a Lipschitz function f on Q is given by the formula.

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 257-263 ◽  
Author(s):  
GERALD BEER ◽  
M. I. GARRIDO
Keyword(s):  
Metric Space ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
The Family ◽  
Locally Lipschitz

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

scholarly journals Flattening functions on flowers

2007 ◽  
Vol 27 (6) ◽  
pp. 1865-1886 ◽  
Author(s):  
EDMUND HARRISS ◽  
OLIVER JENKINSON
Keyword(s):  
Lipschitz Function ◽  
Linear Constraints ◽  
Lipschitz Functions ◽  
Jump Discontinuity ◽  
Connected Components ◽  
Expanding Map ◽  
Maximizing Measure

AbstractLet T be an orientation-preserving Lipschitz expanding map of the circle ${\mathbb T}$. A pre-image selector is a map $\tau : {\mathbb T} \to {\mathbb T}$ with finitely many discontinuities, each of which is a jump discontinuity, and such that τ(x)∈T−1(x) for all $x\in {\mathbb T}$. The closure of the image of a pre-image selector is called a flower and a flower with p connected components is called a p-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given p-flower is shown to be of codimension p in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function f has a maximizing measure S which is Sturmian (i.e. is carried by a 1-flower), it is shown that f can be Lipschitz flattened on some 1-flower carrying S.

Smoothness and Asymptotic Smoothness

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Modulus Of Smoothness ◽  
Lipschitz Functions ◽  
New Approach ◽  
Uniformly Smooth ◽  
Uniform Smoothness ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Frechet Differentiability

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

On the Composition of Differentiable Functions

2003 ◽  
Vol 46 (4) ◽  
pp. 481-494 ◽  
Author(s):  
M. Bachir ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Variational Principle ◽  
Linear Operator ◽  
Differentiable Function ◽  
General Result ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Schur Property ◽  
Differentiable Functions ◽  
Fréchet Differentiable

AbstractWe prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of f ○ T, where f is a Lipschitz function and T is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ‖ ‖lip on various spaces of Lipschitz functions.

ON THE SUPERMODULARITY OF HOMOGENEOUS OLIGOPOLY GAMES

2010 ◽  
Vol 12 (04) ◽  
pp. 309-337 ◽  
Author(s):  
THEO DRIESSEN ◽  
HOLGER MEINHARDT
Keyword(s):  
Characteristic Function ◽  
Demand Function ◽  
Convexity Property ◽  
Profit Function ◽  
Proof Technique ◽  
Net Profit ◽  
Production Technologies ◽  
Similar Proof ◽  
Linear Cost ◽  
Effective Proof

The main purpose is to prove the supermodularity (convexity) property of a cooperative game arising from an economical situation. The underlying oligopoly situation is based on a linear inverse demand function as well as linear cost functions for the participating firms. The characteristic function of the so-called oligopoly game is determined by maximizing, for any cartel of firms, the net profit function over the feasible production levels of the firms in the cartel, taking into account their individual capacities of production and production technologies. The (rather effective) proof of the supermodularity of the characteristic function of the oligopoly game relies on the use of maximizers for the relevant maximization problems. A similar proof technique will be reviewed for a related cooperative oligopoly game arising from a slightly modified oligopoly situation where the production technology of the cartel is determined by the most efficient member firm.

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