Questions tagged [set-valued-analysis]
Tag for questions about set-valued functions and their properties.
75 questions
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Maximal timeinterval for differential inclusions
Let $K \in \mathbb{R}^n$ be a compact (and convex) set and $F(x)$ be an upper semicontinuous set-valued function with compact convex images. I am interested in solutions $x: [0,T] \to K$ to the ...
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Counter-example showing that graph-convexity does not imply lower semicontinuity of set-valued maps
I am trying to find a counterexample showing that graph-convexity does not imply lower semicontinuity of set-valued maps. Since there are many concepts with the same name in the literature, I will ...
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Sequential characterization of upper semi-continuity of set-valued maps
Let $\Lambda: A \rightarrow \mathcal{P}(B)$ be a set-valued map such that $\Lambda(z)$ is a closed set for all $z$, and $A$ and $B$ two metric spaces with $B$ compact. I want to prove that $\Lambda$ ...
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Intersection of Bouligand tangent cones is the tangent cone of the intersection.
I am reading the book "Set-Valued analysis" by Jean-Pierre Aubin and Hèlène Frankowska. On page 152, table 4.4, statement 5b), we read:
If $K_1$ and $K_2$ are closed derivable subsets ...
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About the continuity of a set-valued function that appears in the theory of causal stability from Lorentz geometry.
Edit: Nevermind about the abstract formulation that was here at first, user @MoisheKohan gave an easy counterexample. The question is now reformulated to the actual Lorentz geometry problem at hand.
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Semicontinuity of set-valued maps
Just recently I started working with set-valued maps and things are quite "confusing" and complicated at the moment. So I would like to ask, maybe a silly question, but would appreciate any ...
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Continuity of metric projection with the taxicab metric
Given a compact subset $M$ of $\mathbb{R}^n$, the metric projection associated with $M$ is a function that maps each point $x\in \mathbb{R}^n$ to the nearest points in $M$, that is, $\arg \min_{y\in M}...
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almost single-valued upper hemicontinuous function [closed]
Let $f$ be a set-valued function defined on $\mathbb{R}$.
It satisfies the following conditions:
(1) For any $t \in \mathbb{R}$, $f(t)$ is a closed and bounded subset of $\mathbb{R}$;
(2) $f(q)$ is a ...
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Set valued approximate inversion
I have a function $f: \mathcal{D} \to D$ where $\mathcal{D}$ is some domain of interest.
Now let function $g_\theta: \mathcal{D} \to \mathcal{P}(\mathcal{D})$ be a set valued function, Here $\mathcal{...
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Definition of Random Set
I'm studying the definition of "random closed sets", but I came across two definitions, the first is from Crauel's book and the second I found in Molchanov's book.
$X$ is a Polish space and $...
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Terminology for set-valued functions that are "included" in another function
I was wondering, if you have $f:A \rightarrow B$ a mapping onto sets, is there appropriate terminology for when, for a subset $C \subseteq A$, you have $g:C \rightarrow D$ that verifies $\forall e \in ...
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Clarification between multivalued, set-valued and vector functions
I tried to read as many notes and answers possible before asking this, but I find myself in need of some answer from you guys.
So my confusion is about multivalued functions VS set valued functions VS ...
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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 $U\subset X$, $$
\Psi^{-1}(U):=\{x\,|\,\Psi(x)\cap U\ne\varnothing\}\in\mathcal{B}(X).
$$
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Continuity Properties of Pseudoinverse with Orthogonal Block
Let $p$ be a parameter taking values in a compact set $P$. Let $M(p) = \begin{bmatrix} A(p) & D \\\\ B & C\end{bmatrix}$ be a block matrix. Further impose that $A(p)$ is an orthogonal matrix ...
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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-...
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