Questions tagged [semicontinuity]

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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
3 votes
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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
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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
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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
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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
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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
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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
2 votes
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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
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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
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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
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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
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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
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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
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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
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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
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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