Questions tagged [lipschitz-functions]
For question involving functions satisfying a Lipschitz continuity condition, that is, the distance ratio about the distance of $f(x)$ and $f(y)$ and that of $x$ and $y$ can be bounded independently of $x$ and $y$.
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Proving an integral inequality involving Lipschitz functions
I am trying to prove the following inequality:
$$\int_a^b \Big( f(a)+f(b)-2f(x) \Big)dx\leq \frac{1}{2}\text{Lip}(f) (b-a)^2$$
where $f$ is a Lipschitz function and $a<b$. This inequality is ...
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Different definitions of conformal capacity
A condenser is a pair $(D,E)$, where $D$ is a domain in the plane and $E$ is a compact subset of $D$. The capacity of the condenser $(D,E)$ is defined by:
$$\text{cap}(D,E) = \inf \int_{D} |\nabla u|^...
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Determining if a function is Lipschitz
In a certain exercise I am asked to show that a certain function is Lipschitz. This function is $f(x) = arcsin(x), f:(-1,1) \rightarrow \mathbb{R}$. My attempt consisted on trying to bound the ...
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Reducing a $ n $th order ODE to a system of $ n $ ODEs. A step in the proof
Let $ F $ be a normed vector space, and let $ U_{n - 1},\dots,U_1,U_0\subset F $ be open subsets of $ F $. Let $ I\subset \mathbb R $ be an open interval.
Let $ \mathbf U = U_{n - 1}\times\dots \times ...
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Pointwise Lipschitz and locally Lipschitz
We say that a function $f:\mathbb{R}^n\to\mathbb{R}^m$ is point Lipschitz at $x_0\in\mathbb{R}^n$ if there exists a neighborhood $U$ of $x_0$ and a constant $L>0$ such that:
\begin{align*}
||f(x)-f(...
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Is the divergence of positive definite matrix function locally Lipschitz?
Let $a:\mathbb{R}^d\to S_d^{++}$ where $S_d^{++}$ is the set of $d\times d$ positive definite matrices. Suppose that $a$ is $C^1$. Then by a theorem of Phillips and Sarason (Rogers and Williams ...
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On properties of solution by Euler's method; uniform convergence, etc.
I'm reading about Euler's method to construct approximate solutions to ODEs in Ordinary Differential Equations by Andersson and Böiers. I have questions about properties of the approximate solution. I'...
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Connectedness of the boundary of a domain
I've been struggling to prove the following lemma: "Let $\Omega\subset\mathbb{R}^{d}$ be open and bounded with a Lipschitz boundary and such that $\mathbb{R}^{d}\setminus\partial\Omega$ has ...
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Lipschitz continuous of an integral expression
Let $f(x,y):\mathbb{R}^2\rightarrow[0,+\infty)$ be a continuous function satisfying that:
(1) For any given $x\in \mathbb{R}$, $f(x,y)$ is a probability density function (pdf) of $y$, i.e.
$$\int_{-\...
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Large-scale Lipschitz and bornologous maps
I'm trying to understand the following questions which I took from Roe's book "Lectures on coarse geometry".
A map of metric spaces $f:X \to Y$ is called large-scale Lipschitz if there are ...
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Show that $\frac{e^x}{1+e^x}$ is lipschitz with Lipschitz-constant $\frac{1}{4}$ [closed]
I am trying to show that $\dfrac{e^x}{1+e^x}$ is lipschitz with constant $\dfrac{1}{4}$.
For that I try to show that $\dfrac{e^x}{(1+e^x)^2} \leq \dfrac{1}{4}.$
But I don’t really know how to do this.
...
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How to intuitively view Wasserstein distance dual as moving earth
The Wasserstein-1 distance can be viewed as the minimum amount of work needed to move one distribution to another distribution, as if the distributions were like piles of earth. The typical definition ...
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Where does the gradient of a Lipschitz function end up in?
Let $X = \mathsf{Lip(\mathbb{R}^n;\mathbb{R}^n)}$ be the space of (globally, or perhaps locally) Lipschitz functions, say from $\mathbb{R}^n$ to $\mathbb{R}^n$. From Rademacher's theorem we know that $...
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Lipschitz continuity of function on convex and bounded set; Andersson, Böiers
I'm reading Ordinary Differential Equations by Andersson and Böiers. There is a Lemma regarding Lipschitz continuity which I have some questions about. $\pmb f$ is a vector-valued function, and $\pmb ...
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Lipschitz function in $L^2(\Omega)$.
I read a statement in a paper and I do not think it is correct.
Let $X = L^2(\Omega) \times L^2(\Omega) \times L^2(\Omega)$ where $\Omega \subset \mathbb{R}^2$ is a smooth bounded domain. For $u = (f,...
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Lipschitz constant tending to zero implies function is constant
Let $f$ be a Lipschitz function defined on a ball $B$ in $\mathbb{R}^n$, and assume that the following property on the Lipschitz constant (restricted to smaller balls) holds:
$$
\lim_{r\to 0}\sup_{|x-...
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Is the function $f(x, y)= x\sin y+y\cos x $ Lipschitz on $[-a, a]\times[-b,b]?$
My attempt
Let $(x, y_1)$ and $(x, y_2)$ be two points in the rectangle $R=[-a, a]\times[-b,b].$ Then
$$|f(x, y_1)- f(x, y_2)|\le|x||\sin y_1 -\sin y_2| + |y_1-y_2|$$
I can't understand how to proceed ...
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Separation of normalized molecules in Lipschitz free space
For a pointed metric space $(M,d,0)$ (meaning $(M,d)$ is a metric space and $0\in M$ is a distinguished point), $\text{Lip}_0(M)$ is the Banach space of all real-valued Lipschitz functions $f:M\to \...
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Is the tangent space of a Riemannian manifold a local approximation of the manifold?
Let $(M,g)$ be a Riemannian manifold, $p \in M$. You will often hear people say "$(T_pM, g(p))$ is the infinitesimal geometry of $M$ at $p$". I would like to upgrade `infinitesimal' to '...
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Improve the bound of a Lipschitz function
Let $\lambda$ be a function defined as
$$\lambda(x) = \frac{x}{|\ln (x)|^2}$$
on an interval $[0,x_0]$ for some $x_0 \ll 1$ ($x_0$ can be chose arbitrarily small). Now let us consider $\varepsilon$ ...
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Determining when the function $f(x)=\int_{0}^x g$ is Lipschitz and finding the best Lipschitz constant when it is.
Let $E=(0, \infty)$ and $g \in L^{1}(E)$. Define $f\colon E \to \mathbb{R}$ by $$f(x)=\int_{0}^{x}g.$$
(a) Show that $f$ may not be Lipschitz on $E$.
(b) Prove that if $g \in L^{\infty}(E)$ then $f$ ...
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Show the error squared of the Euclidian projection onto a closed set is a continuous function
I asked this and now I am wondering if the previous proof can be used to show the following:
Let $f(x)=\min_{y \in C}\|y-x\|^2$ where $C$ is a closed set in $\mathbb{R}^n$ and $x,y \in \mathbb{R}^n$.
...
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Hausdorff measure of $f(E)$ is greater than zero where $E$ compact and $f$ is (...)?
I'm studying example 4.1.3 of "Variational Analysis in Sobolev Spaces and BV Spaces" by Attouch, Buttazzo and Michaille. At one point it states that
Given $f: R^n \to R^m \> (n\leq m)$ ...
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Length distance and distance generated by a set of functions.
Let $X$ be a non-empty set and let $A$ be an algebra of bounded functions on $X$ containing the constant functions and separating the points in $X$. Given a distance $\rho$ on $X$, we define $\rho_\...
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Piecewise defined Lipschitz function
Suppose $B \subset \overline{\mathbb{C}}$ is an open set not containing the point at infinity, $v \in C(\overline{B})$ is valued in $[-1,1]$, and Lipschitz on compact subsets of $B$ and $\omega$ is ...
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Growth assumptions on nonlinear term in PDE
In Partial Differential Equations book by Evans they treat a nonlinear system of reaction-diffusion equations. The nonlinearity comes from the reaction term $f$
\begin{align*}
& \partial_t u - \...
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Can I find a lipschitz constant for this term?
I have given two functions $f(x)=\frac{1}{2}x$ and $g(x)=\sqrt{1+x^2}$, $x\in \Bbb{R}$, I want to show that for some $K\geq 0$, $$|f(x)-f(y)|+|g(x)-g(y)|\leq K|x-y|$$ for all $x,y\in \Bbb{R}$. And ...
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Calculating Lipschitz constant of an unknown function from its data
I have an unknown four-dimensional function: $$u:X \subset \mathbb{R}^4 \to Y\, \text{where}\, X := [-5,5] \times [-5,5]\times [-5,5]\times[-5,5],\,Y= [-0.2,0.2],$$ and I have access to its values on $...
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Looking for regularity classes which are weaker than continuity
To contextualize my question, recall that Lipschitz (i.e. $C^{0,1}$) functions by Rademacher's theorem are differentiable almost everywhere. It's interesting to note that this is sharp (and apparently ...
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Check proof that $|f_{a_1}(x)-f_{a_2}(x)| \leq f'(x) |a_1 - a_2|$
Let $a_1, a_2 \in A$ where A is a compact set. Let $x \rightarrow f_a(x)$ be a measurable function such that $a \rightarrow f_a(x)$ is differentiable for almost every $x$. I have to prove that there ...
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Class of Lipschitz Functions on the unit d-dimensional ball
Let $\mathcal{F} = \{f:\mathcal{B}_d \to \mathbb{R}\;:\; \text{f is Lipschitz}\}$, where $\mathcal{B}_d = \{x \in \mathbb{R}^d\;:\: \|x\|_2 \leq 1\}$ is the unit ball in $d$ dimension. Is the class $\...
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Lipschitz functions (Theorem 1.4 of Condenser Capacities and Symmetrization in Geometric Function Theory)
I am struggling to understand the second part of the following proof of Theorem 1.4 of the book "Condenser Capacities and Symmetrization in Geometric Function Theory" by Vladimir N. Dubinin (...
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Sufficient conditions for $f\cdot f^\prime$ to be lipschitz-continuous!
Let $a>0$ (even sufficiently small) and $f:[-a,a]\to\mathbb{R}$ be a Lipschitz-continuous function such that $f^\prime$ is $\alpha$-Hölder continuous with $\alpha<1$ (so not lipschitzian). ...
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K-Lipschitz surjection to higher dimension
I am looking for a (family of) surjective K-Lipschitz function $f : [0,1]^n \rightarrow [0,1]^m$ where $n < m$. I am more interested in an explicit description of such a function than in a proof of ...
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A Lipschitz continuity-style problem [duplicate]
$f$ is a function defined on an interval $I$.
$\exists K\:>0$ such that $\forall x,y\in I$,
$\left|f\left(x\right)-f\left(y\right)\right|\le K\left|x-y\right|^2$.
Show that f is constant on I.
$f$ ...
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Existence and uniqueness of the solution of a control system
Let $T>0$, $(U,d)$ be a metric space and $\mathcal{V}:=\{u:[0,T]\to U\,|\, u\text{ is measurable}\}$. Consider the control system $$\begin{cases}\dot{x}(t)=f(t,x(t),u(t)),\quad \text{a.e. }t\in[0,T]...
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Prove that if $f$ satisfies the Lipschitz condition, then the solutions to $\frac{dx}{dt} = f(x)$ are defined for all $t \in \mathbb R$
Prove that if $f : \mathbb R^n \to \mathbb R^n$ satisfies the Lipschitz condition, then the solutions to $\frac{dx}{dt} = f(x)$ are defined for all $t \in \mathbb R$.
Suppose the function $f$ ...
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Conditions for the squared gradient norm to be convex?
Let $f\colon \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. I am looking for conditions under which the function $$x\mapsto \Vert\nabla f(x)\Vert^2$$ is convex.
It obviously hold if $f$ is ...
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Inequality for Subgaussian Norms and Lipschitz Functions
If I have a $L$-Lipschitz function $\psi \colon \mathbb{R} \longrightarrow \mathbb{R}$ with $\psi(0) = 0$, and a random variable $X \sim \mathcal{N}(0, \sigma^2)$, is it true that
$$ \|\psi(X)\|^2_{\...
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The inverse of an arc length parametrization of $\partial U$ is Lipschitz
Let $U$ be a simply connected open bounded subset of $\mathbb{C}\cong\mathbb{R}^{2}$ with smooth boundary (thus the boundary $\partial U$ is diffeomorphic to a circle). Let $L\geq0$ denote the total ...
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Is a holomorphic $f\colon U\to\mathbb{C}$ with continuous extension to $\overline{U}$ Lipschitz continuous on $\partial U$?
Let $U\subset\mathbb{C}$ be a bounded connected open subset with smooth boundary $\partial U$. Suppose that we have a holomorphic function $f\colon U\to\mathbb{C}$ that can be continuously extended to ...
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Lipschitz Continuity and Lipschitz Constant of a Multidimensional Function
I have a function $f: \mathbb{R}^n \to \mathbb{R}$ defined as follows:
$$
f(x_1,\dots,x_n) = -\frac{x_j-c_j}{(1-k_j)^2}e^{ -\frac{1}{2}\sum_{i=1}^n\left(\frac{x_i-c_i}{1-k_i}\right)^2}
$$
for $j \in \{...
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Lipschitz Hessian implies Lipschitz Hessian diagonal for non-convex function?
I am working on an optimization problem where the function $f$ is assumed to have Lipschitz-continuous gradients and Hessian
\begin{align}
\| \nabla^2 f(x) - \nabla^2f(y) \| \leq L_1 \| x -y \|, \...
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Real analysis, Uniform boundedness and Uniform convergence
Suppose that $f_{n}$be a sequence of real-valued functions that are uniformly Lipschitz and uniformly bounded on $\left[0,1\right]$,there exists constants $K,M>0$ such that for all $n\geq 1$ one ...
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Lipschitz continuous compactly supported function in polar coordinates
Let $f$ be a lipschitz continuous complex valued function in the complex plane, compactly supported in the closed disk of radius $\delta > 0$ with lipschitz constant $M$. Define $h(r,\theta) = f(r \...
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Proving that a function $G: X\rightarrow \mathbb{R}^{n+1}$ is continuous.
$(X, ||\cdot||)$ is a normed vector space, and $l_0, l_1, ..., l_n$ are linear operators in $X^*$. And $G$ is defined like this. $G:X\rightarrow R^{n+1}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x))$. ...
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Lipschitz continuous function | Real Analysis
Let $a, \space b \in \mathbb{R}$ with $a < b$ and consider $f:[a,b] \rightarrow
\mathbb{R}$ s.t $f$ is continuous at $[a, b]$ and Lipschitz continuous
at $(a, b)$. Does it necessarily imply $f$ ...
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Show that in gradient descent the gradient goes against 0 (Under certain conditions)
So we have gradient descent: $$x^{(i+1)} = x^{(i)} - \tau \nabla f(x^{(i)})$$
And we gotta show that $$\left|\nabla f\left(x^{(j)}\right)\right| \to 0$$
The conditions are:
$f: \mathbb R^n \to \...
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Is a Lipschitz continuous function differentiable a.e. for at least one smooth path between every pair of points?
We know by Rademacher's theorem that a Lipschitz-continuous map is differentiable almost everywhere (a.e.).
Now for every path in $\mathbb{R}^n$, $n>1$, there is a Lipschitz-continuous map such ...
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Can I smooth a map to the circle, while preserving the Lipschitz constant?
Let $M$ be a closed Riemannian manifold (perhaps even a surface, at first), and let $u: M \to \mathbf S^1$ be a Lipschitz map to the circle, with Lipschitz constant $L$. We can find an approximation $(...
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