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Are hyperplanes Upper and Lower semicontinous correspondence? [closed]

Let $\mathcal{\Phi}:\mathbb{R}^2 \to P(\mathbb{R}^2)$, where $P(\cdot)$ is a power set, defined by $\mathcal{\Phi}(d) = \{x\in \mathbb{R}^2: d^Tx=0\}$ where $d^T$ is the transpose of vector $d$. So, ...
Uday Kumar's user avatar
1 vote
0 answers
75 views

Sum of upper semi continuous and lower semi continuous functions

Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
Adam's user avatar
  • 1,011
1 vote
2 answers
168 views

Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is. ...
Anacardium's user avatar
1 vote
1 answer
124 views

On the additive property of the subdifferential of lower semicontinuous functions

Let $f:\mathbb R\to\mathbb R$ be a lower semicontinuous function, we define the Fréchet subdifferential of $f$ at $x\in\mathbb R$ by $$\partial^F f(x):=\left\{L\in\mathbb R: \liminf_{v\to0}\frac{f(x+v)...
Fergns Qian's user avatar
1 vote
0 answers
21 views

Weakening compacity hypothesis in multifunctions intersection

Let $X,Y$ be metric spaces, $x^*\in X$ We define two multifunctions $F_1:X\rightrightarrows Y$,$F_2:X\rightrightarrows Y$. We recall the upper-semi-continuity in Berge's sense : A multifunction $F:X\...
Hamdiken's user avatar
  • 141
3 votes
0 answers
194 views

Semicontinuity of Exts

Let $Z$ be a proper noetherian scheme over an algebraically closed field $k$, and $S$ a smooth scheme over $k$. Let $\cal E$, $\cal F$ be coherent sheaves on $Z\times S$, flat over $S$. Is it true ...
Hephaistos's user avatar
1 vote
1 answer
152 views

Right-continuity of covering number

Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some ...
iom10's user avatar
  • 23
2 votes
1 answer
112 views

A question about a realcompact space and upper semicontinuous function

Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative upper ...
Mehmet Onat's user avatar
  • 1,211
2 votes
2 answers
535 views

A net of lower semicontinuous functions

Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise. Please is it true that one can extract a countable ...
Oleg Zubelewicz's user avatar
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 $$ d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0. $$ ...
ABIM's user avatar
  • 4,775
0 votes
0 answers
118 views

From convergence pointwise to convergence of the supremum for semicontinuous functions

Let $K\subset\mathbb{R}$ a compact set, and $(f_n)_{n\geq 1}$ and $f$ upper semicontinuous functions over $K$ (taking hence values in $\mathbb{R}\cup\{-\infty,+\infty\}$) such that for all $x\in K$, ...
G. Panel's user avatar
  • 619
2 votes
1 answer
298 views

Semi-continuity in quasi-finite morphisms without properness

Let $f:U \to V$ be a flat, quasi-finite, surjective morphism between two affine varieties defined over $\mathbb{C}$. Assume that every closed fiber is reduced. Consider the function $\eta$ that sends ...
Chen's user avatar
  • 1,583
1 vote
1 answer
61 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}^{...
Marko Rajkovic's user avatar
3 votes
1 answer
686 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 ...
iolo's user avatar
  • 621
2 votes
3 answers
457 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 ...
ECL's user avatar
  • 281
0 votes
0 answers
268 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 ...
user267839's user avatar
  • 6,034
1 vote
0 answers
61 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 ...
CuriousTiger's user avatar
0 votes
1 answer
132 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}...
ABIM's user avatar
  • 4,775
2 votes
0 answers
117 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 \...
Raffaele C's user avatar
2 votes
0 answers
37 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 ...
ABIM's user avatar
  • 4,775
2 votes
1 answer
322 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 ...
Mehmet Onat's user avatar
  • 1,211
2 votes
1 answer
133 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=...
ABIM's user avatar
  • 4,775
1 vote
2 answers
116 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 ...
amateur's user avatar
  • 375
2 votes
1 answer
150 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-...
user429197's user avatar
2 votes
1 answer
182 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} \...
StopUsingFacebook's user avatar
0 votes
0 answers
125 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 $\...
R. N. Marley's user avatar
0 votes
2 answers
333 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. ...
Adam's user avatar
  • 1,011
5 votes
1 answer
2k 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: ...
e.lipnowski's user avatar
3 votes
1 answer
602 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 ...
vicubso's user avatar
  • 131
0 votes
1 answer
90 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(...
M. Reza. K's user avatar
2 votes
1 answer
166 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 ...
M. Reza. K's user avatar
0 votes
1 answer
258 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 ...
M. Reza. K's user avatar
4 votes
1 answer
249 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 ...
Matteo Triossi's user avatar
2 votes
0 answers
81 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 ...
Martha Łącka's user avatar
1 vote
1 answer
151 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}\...
Idonknow's user avatar
  • 603
2 votes
0 answers
483 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 ...
Idonknow's user avatar
  • 603
1 vote
1 answer
1k 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 ...
Xifeng Su's user avatar
  • 173
1 vote
0 answers
34 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 \...
Vladimir's user avatar
  • 1,200
7 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 ...
flyingwith's user avatar
2 votes
1 answer
113 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's user avatar
  • 201
1 vote
0 answers
967 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's user avatar
  • 201
6 votes
1 answer
507 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) - \...
Gregor Kemper's user avatar
8 votes
1 answer
243 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,\...
p. duarte's user avatar
1 vote
0 answers
66 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 ...
user70457's user avatar
8 votes
0 answers
525 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 ...
David E Speyer's user avatar
12 votes
2 answers
733 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 ...
alvarezpaiva's user avatar
  • 13.2k
14 votes
6 answers
5k views

More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry. Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...