Questions tagged [set-valued-analysis]

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3
votes
2answers
35 views

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. ...
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0answers
16 views

Graphical Convergence

Suppose that $U$ is a dense subset of $\mathbb{R}^d$, what is an example of a lower semi-continuous map $f:\mathbb{R}^d\rightarrow \mathbb{R}$, for which $f|_U$ converges graphically to $f$? Note: ...
3
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1answer
53 views

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}^...
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1answer
38 views

On the intersection of Bouligand cones

Let $K, L$ be two closed convex subsets of the normed space $X$. If $$ 0 \in \operatorname{Int}(K-L) $$ Prove that $$ \forall x \in K \cap L \Rightarrow T_{L \cap K}(x)=T_{K}(x) \cap T_{L}(x) $$ In ...
1
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1answer
50 views

fixed point of a multivalued map

Suppose I have a map $S:X \to Y$ between Banach spaces which is multivalued, so that $S(x)$ is a set. I have shown that $S$ takes a closed ball of radius $R$ to itself, and it also is such that if $...
2
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1answer
74 views

Measurable Projection to product space

Let $(\Omega,\mathcal{F},P)$ be a complete probability space; $X, Y$ complete separable metric spaces. The Measurable Projection Theorem says that if the set $G\in\mathcal{F}\otimes\mathcal{B}(Y)$ ...
1
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0answers
55 views

About measurablility of Caratheodory maps

Let $(\Omega,\mathcal{F})$ be a measurable space and $X, Y$ be separable Banach space. Consider a Caratheodory mapping $\varphi:\Omega\times X\to Y$, i.e $\forall x\in X$, $\varphi(\cdot,x)$ is ...
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0answers
31 views

Weakly closed cone-valued map

I am looking for a non-trivial set-valued map with the following properties: $K:X \rightrightarrows Y$ is a weakly closed set-valued cone-map, where for every $x\in X$, $K(x)$ is proper, closed, ...
1
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1answer
50 views

The smoothness of a set-value function (correspondence)

I learned the upper-hemicontinuity and L-continuity of set-valued function. Are there a definition of smoothness or differentiability of a set-valued function $$f:\mathbb R\rightrightarrows\mathbb ...
0
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1answer
56 views

How can I prove a closed ball is closed on metric topology?

Let $(X,\tau _d)$ is metric topology. Show that $B_r[x]$ is closed $X\setminus B_r[x]$ must be open set. It is open iff $X\setminus B_r[x] \in \tau _d$ iff $\forall y \in X\setminus B_r[x] , \...
2
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1answer
140 views

Set-valued maps: locally Lipschitz implies upper semicontinuity

With $X$ and $Y$ metric spaces, let $F:X\rightrightarrows Y$ be a set-valued map satisfying the following (P) For each compact set $K\subset \operatorname{dom} F$ there exists $L>0$ such that, ...
2
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1answer
88 views

Are single-valued function also set-valued functions?

Consider the single-valued functions $f: \mathbb{R}^n \to \mathbb{R}$, and the set valued function $F: \mathbb{R}^n \to 2^{\Bbb{R}}$. Is $f$ a set-valued function as well? My only trouble in ...
2
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1answer
242 views

Prove that oscillation function is upper semicontinous

Suppose $X$ is a Polish space and $Y$ a compact metrizable space. Denote $P_k(Y)$ to be the collection of all non-empty, compact subsets of $X$. Denote the Hausdorff metric $d_H : P_k(Y) \rightarrow ...
1
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1answer
57 views

Multivalued function: lower semicontinuity equivalent condition.

I have to prove that: $$ F\text{ - l.s.c. in } x_0\Rightarrow\forall (x_n)_n\subseteq X\text{ s.t. }x_n\rightarrow x_0,\forall y_0\in F(x_0),\exists(y_n)_n\subseteq Y \text{ s.t. } y_n\rightarrow y_0 \...
0
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0answers
85 views

Why is $(A - \lambda I)^{-k}$ closed when a convex process $A$ is closed?

Let $A$ be a closed convex process from $R^n$ to $R^n$, $I$ be the identity map, $\lambda$ be a real number, and $k$ be a positive integer. It is obvious that $A - \lambda I$ is closed and so is $(A - ...
2
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1answer
265 views

Lipschitz continuity in product metric spaces in terms of (certain) marginal sets

Let $(X,d_X)$ and $(Y,d_Y)$ be compact metric spaces ($X,Y$ with at least two points each to avoid triviality) and $f:X\times Y\to\mathbb{R}$ a Lipschitz function, that is, there is $C>0$ such that ...
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0answers
163 views

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 $x_0\in X$ ...
1
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3answers
296 views

Corollary of Tietze extension theorem

The Tietze extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $g:...
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0answers
109 views

How to Compute or Find an Upper Bound for the Diameter of a Convex Set?

Let $\mathcal{R}$ be an $n$-dimensional bounded hyperrectangle, and consider a $n\times n$ matrix $A$ with real entries. Given set $\mathcal{R}$ and matrix $A$, I want to compute or find an ...
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0answers
119 views

convex hull of a lower hemicontinuous correspondence is lower hemicontinuous

Let $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ be two topological spaces. We call a correspondence (set-valued function) $F: X \rightarrow 2^Y$ lower semicontinuous if for every $G \in \mathcal{T}'$ ...
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0answers
174 views

Multilinear extension of submodular function

I am reading the wikipedia article about submodular functions. Let $\Omega$ be a finite set and $f\colon 2^\Omega\to \Bbb R$ a submodular set function, i.e. a function such that $$f(S)+f(T)\geq f(S\...
2
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0answers
174 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
2
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0answers
116 views

Definition of closed multivalued operator.

I have a question about closed operator. Operator $F:X \to Y$ is closed if $(x_n)$ is a sequence in $D(F)$ that is convergent in $X$ and the sequence $(F(x_n))$ is convergent in $Y$, then we have $\...
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0answers
146 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
1
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1answer
220 views

Lower and upper semicontinuity of the Cartesian product

Suppose I am considering set-valued maps $G_i:\Bbb R\to 2^\Bbb R$ which I know are both upper and lower continuous. Does it mean that the product map $G_1\times\dots\times G_n$ is upper and lower ...
0
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1answer
218 views

uniformly bounded on compact set

Let $F:\Omega\to 2^X\setminus\emptyset$, $\Omega$ is compact. If $F$ has bounded values then they are uniformly bounded. Please help me prove this! Thanks!
4
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0answers
236 views

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 ...