Mathematics

Questions tagged [set-valued-analysis]

Tag for questions about set-valued functions and their properties.

75 questions
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Monotone (measurable) selection theorem

Given a Polish space $X$ and a set valued map $\Psi:X\to 2^X$ we way $\Psi$ is weakly Borel if for every open $U\subset X$ , $$ \Psi^{-1}(U):=\{x\,|\,\Psi(x)\cap U\ne\varnothing\}\in\mathcal{B}(X). $$ With the additional assumption that $\Psi(x)$ is…
APP
  • 439
6
votes
1 answer

Measurable argmax correspondence on probability spaces

I'm trying to wrap my head around the "Measurable maximum theorem", Thm 14.91 in "A hitchhiker's guide to infinite dimensional analysis" by Aliprantis & Border. I wonder if I can use it in a case when the underlying measurable space is a probability…
user202542
  • 941
5
votes
1 answer

Prove the (path-) connectedness of the graph of a compact- and convex-valued upper hemi-continuous correspondence.

Let $\Gamma: [0, 1] \to \mathbb{R}$ be a compact- and convex-valued, upper hemi-continuous correspondence. Prove that the graph of $\Gamma$ is a connected set. Is it path-connected? This is what I have so far: Proof $\space$ We first show that…
Beerus
  • 2,969
5
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algebraic or homotopical proof for Kakutani fixed point theorem

As Kakutani fixed point theorem is a genral case of Brouwer fixed point theorem, and one can read the proof from homotopy theory books. I wonder if there is any proof for the Kakutani using homotopy theory or if it has at least some applications in…
Rahman
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4
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A question on the directional derivative in a real Banach space

Let $K$ be a convex and compact set in the real Banach space $(\mathbb{R}^n, |\cdot|_{\infty})$ . Here, the max-norm is defined as $|v|_{\infty}=\max_{i}|v_{i}|$ and the distance function $d_{K}(\cdot)$ is defined to be $d_{K}(x):=\inf_{y \in…
HTChoi
  • 105
4
votes
1 answer

Subdifferential of a proper convex function: is it upper-hemicontinuous?

I am fairly new with the concept of upper-hemicontinuity, i.e. Let $X \subseteq E^{n}$ , $Y \subseteq E^{m}$ and $\Psi: X \rightrightarrows Y$ be a set-valued map. $\Psi$ is upper hemicontinuous at $x_{0} \in X$ if, for every open set $V \supseteq…
Julien V
  • 165
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References about the historical move from set-valued maps to single-valued maps, and from curves to functions.

In Aubin, Jean-Pierre, and Hélène Frankowska. Set-valued analysis. Springer Science & Business Media, 2009. We read that historically, set-valued analysis was being developed by Kuratowski and Hausdorff, along with other Polish and French…
jadn
  • 469
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Question about the equivalence of three versions of Closed Graph Theorem

I am studying set-valued analysis, and I saw three version of this Closed Graph Theorem. Version 1: (what I was taught in class) Let $\Gamma: X \to Y$ be a correspondence. If $\Gamma$ is closed-valued and upper hemi-continuous, then $Gr(\Gamma)$ is…
Beerus
  • 2,969
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Continuity of a set-valued function

Let $f:\mathbb{R}^{n}\rightrightarrows \mathbb{R}^{m}$ be a set-valued function defined by \begin{equation*} f\left( x\right) =\left\{ y\in \mathbb{R}^{m}:g\left( x\right) +h\left( x\right) ^{T}y\leq 0\right\} \text{,} \end{equation*} where…
UnclePetros
  • 63
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On the continuity of the funciton $w \mapsto \mu(\{x \in X \mid w^\top v(x) > t\})$

Let $X$ be a measurable space and let $\mu$ be a probability measure on $X$ (assumed to be non-atomic , in case that helps). Let $v:X \to \mathbb R^k$ be a $\mu$ -integrable function and for every $t \in [0,b]$ (with $b>0$ fixed), consider the…
dohmatob
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Differntial inclusions that are single valued near hyperbolic rest point: Hartman Grobman still fine?

Suppose for $x\in \mathbb{R}^m$ we have a well behaved differential inclusion $\dot x \in F(x)$ , where $F(x)$ is a Marchaud map, i.e. u.s.c. with compact convex values and linear growth condition: there exists $c>0$ s.t. $$\sup\{\|y\|:\, y\in F(x)…
Clempe
  • 61
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2 answers

Does the closure of a strict sublevel set of a continuous function equal a non-strict sublevel set?

Let $X \subset \mathbb{R}^{n}$ be closed and $f$ be a continuous real valued function on $X$ . Consider now the sets \begin{align} S_{<} = \{ x \in X : f(x) < 0 \}, && S_{\le} = \{ x \in X : f(x) \le 0 \}. \end{align} Is it generally true that if $…
node
  • 203
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Passing to weak-strong limit in pointwise inclusions

Let $F:\mathbb R^m\rightrightarrows\mathbb R^n$ be a set-valued map (or multi-function, correspondence) with $F(x)\ne\emptyset$ for all $x\in \mathbb R^m$ . Let $I\subset\mathbb R$ be an interval. Let be sequences of functions $(y_n)$ and $(x_n)$ in…
daw
  • 55,743
3
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1 answer

Graph of the composition of a continuous function and a correspondence with a graph homeomorphic to its domain

Let $\phi: \mathbb{R}^m \leadsto \mathbb{R}^n $ be an upper hemi-continuous correspondence , $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous function. If the graph of $\phi$ , $\{(x,y) \in \mathbb{R}^{m+n} \mid y \in \phi(x) \}$ , is homeomorphic…
Metta World Peace
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Is an upper semicontinuous correspondence weakly measurable?

Definition 1 Let $F:X\to2^Y$ be a set-valued map from a metric space to the subsets of another metric space. We say it is upper semi-continuous (USC) if for every $\epsilon$ and every $x_0\in X$ there exists $\delta$ such that…
MickG
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