# Questions tagged [set-valued-analysis]

Questions about maps into the powerset of a set (called set-valued maps, multivalued maps, or relations), corresponding concepts of continuity (like upper and lower semicontinuity), inclusion problems (like differential inclusions), maximal monotone maps, hyperspaces (families of subsets of a set, endowed with the Hausdorff distance or Vietoris topology), etc.

34
questions

4
votes

2
answers

182
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### Minimal norm of Fréchet subdifferential for function Lipschitz over its domain

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{+\infty\}$ be an extended real-valued function that is proper, lower semicontinuous, and Lipschitz continuous over its domain $\newcommand{\dom}{\text{dom}...

1
vote

1
answer

103
views

### Do Gromov hyperbolic spaces admit concical geodesic bicombings?

Consider a metric space $(X,d)$ with a distinguished selection of geodesics, i.e. a geodesic bicombing $\sigma:X\times X\times [0,1]\rightarrow X$. We call a geodesic bicombing conical if it ...

5
votes

1
answer

232
views

### On the continuity of a Set-Valued function (correspondence) [closed]

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\...

2
votes

0
answers

63
views

### Conditions ensuring that a compact set can be approximated by approximating its distance function

Let $\emptyset\neq K\subset Y$ be a closed subset of a compact metric space $(X,d)$ such that $K$ has at-least two points and such that
$$
d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0.
$$
...

0
votes

0
answers

108
views

### Finite sets are residual in the Hausdorff space

Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(...

2
votes

0
answers

73
views

### Convex optimization over compact sets defined as Aumann set-valued integrals

Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is:
closed (i.e $F(x)$ is closed for ...

1
vote

0
answers

72
views

### On the closedness of a certain subset of $\mathbb R$

Let $\mu$ be a probability measure on measurable space $X=\mathbb R^n$ (euclidean), and let $F$ be a family of $\mu$-measurable functions $X \mapsto \mathbb R$ which are uniformly bounded, i.e $b:=\...

2
votes

1
answer

94
views

### Compactness of the integral of a set-valued function

Let $X$ be a compact space (e.g. a compact subset of $\mathbb R^n$) and $P$ be a probability measure on $X$. Let $A$ be a compact subset of some $\mathbb R^d$. Finally, let $F$ be the collection of $P$...

1
vote

3
answers

339
views

### Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?

Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...

0
votes

1
answer

222
views

### Existence of a global solution to a differential inclusion that does not blow up

Let $\dot{x}(t) \in F(x(t))$ be a differential inclusion, with $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ an uppersemicontinuous, convex and compact valued set-valued map.
On Wikipedia it is ...

9
votes

1
answer

239
views

### Homotopy type of the Hausdorff metric

Recall that if we have a metric space $X$ then we can consider the set of its nonempty compact subsets and equip this with a metric called the Hausdorff distance. Denote the resulting metric space $\...

0
votes

1
answer

229
views

### Continuity of Kakutani fixed points

Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By ...

0
votes

1
answer

102
views

### Mapping problem reminiscent of Mastermind

Given 2 finite sets $S$ and $M$, with $\operatorname{card}(S) \geq \operatorname{card}(M)$, and an item $z \not\in M$. There is an unknown function $f: S \to M \cup \{z\}$, which is known to be one-to-...

4
votes

2
answers

243
views

### Tangent cone of null sets

Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that
$$ \liminf_{r \to 0+} r^{-1}...

4
votes

0
answers

108
views

### "Snowflaked" Hausdorff metric

Let $(X,d_X)$ be a compact metric space and let $Comp(X)$ be the set of closed subsets of $X$ with the Hausdorff metric:
$$
D(A,B)\overset{\text{def}}{=} \, \max\left\{\sup_{b\in B}\,d_{A}(b),\sup_{a\...

0
votes

0
answers

81
views

### Gromov–Hausdorff closure of non-positively curved graphs

Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. ...

12
votes

1
answer

1k
views

### Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$.
Suppose that:
$K$ is regular : ...

4
votes

0
answers

292
views

### Relationship between Hausdorff convergence of sets and indicator functions

Let $\{K_n\}_n$ be a sequence of compact subsets of a metric space $X$, and $K\subset X$ be compact. If $K_n$ Hausdorff converges to $K$, i.e.:
$$
\lim\limits_{n\to\infty} d_{\mathrm H}(K_n,K) = \max\...

5
votes

1
answer

182
views

### Criterion for Kuratowski Limit Inferior

Let $(X,d_X)$ be a compact metric space and let $\{K_n\}_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x_n \in K_n$ we have
$$
d_X(...

1
vote

1
answer

129
views

### Conditions for pointwise convergence of indicators precomposed with uniformly continuous sequence

Let $X$ be a compact metric space, $\{\delta_n\}_{n=1}^{\infty}$ be a strictly monotonically decreasing sequence in $[0,1]$ converging to $0$, and $\{h_n\}_{n=1}^{\infty}$ be a uniformly convergence ...

1
vote

0
answers

107
views

### Is the Vietoris topology on compact subsets of $\mathbb R^n$ locally convex?

The title question says it all really.
If the question is negative for compact subsets of $\mathbb R^n$, is it affirmative for compact and convex subsets of $\mathbb R^n$? How about for all nonempty ...

9
votes

0
answers

252
views

### Does a generalization of Tietze's extension theorem hold for set-valued functions?

Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...

3
votes

0
answers

67
views

### Is there a complex Katetov-Tong theorem?

For any bounded function $f: X \to \mathbb R$, not-necessarilly continuous, one can define for any $x$, the real functions
$$
\limsup f(x) = \inf_{U\in\mathcal V_x} \sup_{u\in U} f(u)
$$
and
$$
\...

5
votes

1
answer

274
views

### 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{...

0
votes

2
answers

165
views

### Intersection with a fixed set in Hausdorff metric space [closed]

Let $\mathbb{R}^d$ be a the usual Euclidean space and let $Y$ be a fixed non-empty closed subset of $Ball(0,1)$ (the unit ball in $\mathbb{R}^d$ about $0$ of radius $1$).
Let $f$ be the map taking $...

0
votes

1
answer

92
views

### Topology on closed subsets characterized by sup on continuous functions?

Let $(X,d)$ be a metric space. Suppose that $\{A^n\}_{n \in \mathbb{N}}$ is a sequence of closed, non-empty subsets of $X$.
Is there a Hausdorff topology on the space of closed subsets of $X$, ...

2
votes

0
answers

299
views

### Lipschitz min implies Lipschitzian argmin?

Let $X$ be a Hilbert space, and suppose that $f:X^2\rightarrow \mathbb{R}$ is a Lipschitz, supercoercive, convex function such that (for every $y \in X$) the set
$$
\operatorname*{argmin}_{x\in X} f(x,...

2
votes

0
answers

157
views

### Reference Request: A Set-Valued Minimax Theorem?

Suppose that $\mathcal{C}$ and $\mathcal{D}$ are subsets of $L^2(X,\Sigma,\mu)\cap L^{\infty}(X,\Sigma,\mu)$, where $\mu$ is a finite-measure on $(X,\Sigma)$. Let $F:L^2(X,\Sigma,\mu)\times L^2(X,\...

1
vote

0
answers

71
views

### Closeness of the product of closed convex processes

I asked this question to the math.stackexchange but couldn't get an answer.
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 ...

6
votes

2
answers

3k
views

### Upper semicontinuity of set-valued maps with open values

Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as:
Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at ...

3
votes

2
answers

346
views

### Random processes with smooth paths

Is there any prototypical example of a Random process with smooth paths?
I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path.
It's easy to ...

3
votes

0
answers

193
views

### Is there a name for this property in set-valued analysis?

Consider a set-valued, finite-valued map $F$ from a set $X$ to subsets of $X$. Consider the following property: $|F(x)| \geq |F(y)|$ for all $x,y$ such that $y \in F(x)$.
I have defined this property ...

2
votes

0
answers

172
views

### Differential inclusions for distributions

Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$.
I was wondering if someone ...

5
votes

1
answer

556
views

### Do upper-semicontinuous polyhedral multifunctions have Lipschitz continuous selections?

We are interested in the following question (definitions and references are given below):
Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there ...