# Questions tagged [set-valued-analysis]

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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. ...
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### 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: ...
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### 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 $G\in\mathcal{F}\otimes\mathcal{B}(Y)$ ...
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### 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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### 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, ...
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### 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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### 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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### 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\...
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### 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 ...