Questions tagged [semicontinuity]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
45 views

Weak lower semicontinuity of a sequence of Riemann sums

Let us have a sequence of functions $\{f^K\}_{K \in \mathbb{N}} \in C([0,1],\mathbb{R})$ which is uniformly bounded in $L^2((0,1))$. We observe a sequence of Riemann sums $$R^K=\frac{1}{K} \sum_{k=0}^{...
user avatar
1 vote
1 answer
127 views

Is a convex, lower semicontinuous function that is bounded from below, actually continuous?

While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space. Lemma: Let $f ...
user avatar
  • 509
1 vote
3 answers
135 views

Looking for a reference: $f$-divergences are lower semicontinuous

I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability ...
user avatar
  • 231
0 votes
0 answers
229 views

Flatness of affine cone due to semicontinuity theorem

I would like to clarify an important aspect from the discussion in this question. The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic Geometry Chap. III page ...
user avatar
  • 1,015
1 vote
0 answers
48 views

Lower semicontinuity of the number of attracting periodic points of a holomorphic family of rational maps?

Recently I have been reading the book Mathematical Tools for One-Dimensional Dynamics. In the proof of the theorem 5.4.2, authors use the following fact that the number of attracting periodic points ...
user avatar
0 votes
1 answer
93 views

Lower semi-continuity of induced function on sequences

Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum_{n=1}^{\infty}...
user avatar
  • 5,008
2 votes
0 answers
78 views

Semicontinuity of length for coherent sheaves

Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \...
user avatar
2 votes
0 answers
28 views

Weak relaxation of a strongly lower semi-continuous functional

Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
user avatar
  • 5,008
2 votes
1 answer
233 views

Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is defined as follows: $ f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in U\right\} :U\in ...
user avatar
2 votes
1 answer
80 views

Lower semi-continuity of length-dependent functional

Let $f:\mathbb{R}\rightarrow [0,\infty]$ be a lower semi-continuous function and define the functional $$ \begin{aligned} F_f:&\ell^1 \rightarrow [0,\infty]\\ (x_n)_{n=0}^{\infty} &\to \sum_{n=...
user avatar
  • 5,008
1 vote
2 answers
100 views

Rank of a linear combination of linear operators

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow. Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now ...
user avatar
  • 355
2 votes
1 answer
143 views

Regarding upper semicontinuity of a function

Let $E$ be a linear subspace of $\mathbb{C}^{n\times n}$. Define the function $\mu_E:\mathbb{C}^{n\times n}\longrightarrow \mathbb{R}_+$ as $$ \mu_E(A)=\frac{1}{\inf\{\|X\|:X\in E\text{ and }\det(I_n-...
user avatar
2 votes
1 answer
66 views

Weak lower semicontinuity of functional with two arguments

Let $\Omega$ be a bounded domain (smooth if necessary) and let $J:H^1(\Omega) \times H^1_0(\Omega) \to \mathbb{R}$ be defined by $$J(u,v) = \int_\Omega f(u)|\nabla v|^2$$ where $f\colon \mathbb{R} \...
user avatar
0 votes
0 answers
88 views

Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
user avatar
0 votes
2 answers
252 views

Goldowsky-Tonelli theorem for upper semi continuous function

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...
user avatar
  • 803
5 votes
1 answer
1k views

When do convexity and lower semicontinuity imply continuity?

Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous. Question: ...
user avatar
3 votes
0 answers
336 views

Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes: Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...) for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of ...
user avatar
  • 131
0 votes
1 answer
87 views

If a function follows another one's range order, can we say it follows some continuity properties?

Suppose that $(X,d)$ be a complete metric space, $f:X \to \mathbb{R}$ is a sequentially lower monotone function. Let $g: X \to \mathbb{R}$ be a function with the property: $f(x)\leq f(y) \Rightarrow g(...
user avatar
1 vote
1 answer
100 views

Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...
user avatar
0 votes
1 answer
186 views

An extension for lower semi continuous lower bounded real valued functions class

Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
user avatar
4 votes
1 answer
190 views

Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...
user avatar
2 votes
0 answers
79 views

lower semicontinuity of the number of extreme points

Do you know the reference for the following fact: the number of extreme points of a compact convex subset of a locally compact space varies lower semicontinuously when we endow the space of compact ...
user avatar
1 vote
1 answer
119 views

Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?

Ian Morris quoted the following: For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
user avatar
  • 593
2 votes
0 answers
355 views

When will the upper regularization of a bounded function not defined?

Suppose $E$ is a compact metric space. A function $f :E \rightarrow \mathbb{R}$ is upper semicontinous if for all $c \in \mathbb{R}$, $f^{-1}(-\infty, c)$ is open in $E.$ For any real-valued ...
user avatar
  • 593
1 vote
1 answer
855 views

properties of orderd upper and lower semi continuous functions [closed]

$M$ is a compact space. Assume $f$ is upper semi-continuous on $M$, $g$ is lower semi-continuous on $M$, and $f(x) \geq g(x)$ for any $x\in M$. If $f(x_0) = g(x_0) $ for some point $x_0\in M$, is it ...
user avatar
  • 173
1 vote
0 answers
28 views

Semi-continuity extends to dual

Say $f \colon X \to \mathbb{R}$ is a lower semi-continuous on a compact space $X$. Let $\mathcal{P}(X)$ denote the space of Borel probability measures on $X$, and let $f^* \colon \mathcal{P}(X) \to \...
user avatar
  • 1,270
6 votes
2 answers
2k 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 ...
user avatar
2 votes
1 answer
110 views

Lebesgue measurability of singular set

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function. Define a superdifferential of $f$ at $x\in Q$ by $$ D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid \text{$f(y)\...
user avatar
  • 201
1 vote
0 answers
768 views

Approximation of semicontinuous functions by continuous (or smooth) functions with closed form

I'm looking for a sequence $(f_{\epsilon})_{\epsilon>0}$ of continuous (or smooth) functions approximating a semicontinuous function $f$. Here, for approximation, pointwise convergence is fine. For ...
user avatar
  • 201
6 votes
1 answer
425 views

Is the embedding dimension minus the dimension upper semicontinuous?

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - \...
user avatar
8 votes
1 answer
235 views

Do semialgebraic sets depend outer semicontinuously on their defining polynomials?

Consider a compact (semialgebraic) ball $B\subset \mathbb{R}^n$ and a semialgebraic set $A=A(f_1,\cdots,f_s)\subset \mathbb{R}^n$ defined through some representation in terms of polynomials $f_1,\...
user avatar
1 vote
0 answers
65 views

Upper semicontinuity in C(X)-algebras. Quotient norm question

upper semicontinuity in C(X)-algebras In the 5th paragraph of this post, I don't understand why there exists a vector b satisfying $||a+b||_A < ||q_x (a)||_A(x)$ By the definition of the quotient ...
user avatar
7 votes
0 answers
464 views

Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
user avatar
11 votes
2 answers
650 views

Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself: Definition. Let $(X,d)$ be a ...
user avatar