# Questions tagged [set-valued-analysis]

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### How to define Random Set (set-valued RV) for closed sets (non real values)?

I'm trying to define a set-valued Random Variable (RV) whose set values are not real points. Instead, I'm trying to define something more general: Definition (Random Set): A Random Set is a set-valued ...
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### Is it possible to define the notion of a Submanifold of Euclidean space through properties of its tangent cone?

The tangent cone to a set $\mathcal S \subset\Bbb R^n$ at a point $x \in \Bbb R^n$ is the set of all vectors $w \in \Bbb R^n$ for which there exists sequences $x_i \in \mathcal S$ and $\tau_i> 0$, ...
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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 ...
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### 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 \...
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### Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{...
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### Interchange Argmin and Monotone Function

Suppose that $f:X\rightarrow (0,\infty)$ is lower semi-continuous, and coercive (so it admits a minimizer) and suppose that $g:(0,\infty)\rightarrow (0,\infty)$ is monotone increasing and smooth. ...
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### How to show upper/lower-hemicontinuity of a correspondence - with non-hand-wavy arguments?

I understand (at least I think I understand) the notion behind these topics. In my class we use the following criterion: A correspondence $\Gamma: X \rightrightarrows Y$ is LHC at $x$ if whenever $x$...
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### Difference between normalizing and averaging with probability sum rule

Given a joint distribution $X,Y$, I'm trying to get the probability of $X=x$ using the sum rule: $$P(x)=\int f(x,y)dy.$$ I undestand that I need to solve for the normalizing constant $K$, to obtain a ...
64 views

### How to show that a set-valued function is lower semicontinuous?

Let me recall the definitions of lower and upper semicontinuity of real-valued functions. Definition 1. Let $X$ be a topological space and $f:X→\mathbb R$. We say that $f$ is lower semicontinuous (...
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### 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 ...
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### 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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### 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 $x_0\in X$ ...
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### 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 ...
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### 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 ...
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!