# Tagged Questions

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### Show homeomorphism between convex hull and unit ball?

In the proof of Schauder fixed point theorem in Evans' PDE book, he uses a claim that the convex hull $K$ of $N$ points $x_1,\dots,x_N$ in a convex compact subset $A$ of a Banach space $X$ is ...
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### How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
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### Why is the projection of a closed polytope closed?

In general, projection of a closed set into a subspace does not result in a closed set. However, I was able to prove that in $\mathbb{R}^n$, the projection of a closed polytope (intersection of ...
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### Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
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### closed epigraphs equivalence

Is there a way to prove that the epigraph of any real function $f$ is closed iff $f$ is lower semi-continuous without using limit superior or inferior?
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### A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
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### interior points and convexity

Question: Let If $P\subseteq \mathbb{R}^n$ be a convex set. show that $int(P)$ is a convex set. I know that a point $x$ is said to be the interior point of the set $P$ if there is an open ball ...
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### Generalizing the Definition of Convexity

The definition of convexity can be given as: Definition: Call a subset of $\mathbb{R} ^ k$, which will be denoted $E$, convex if given two elements of $E$, $\boldsymbol{x}$ and $\boldsymbol{y}$ and ...
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### Interior point and Minkowski functional

I want to prove for a convex set that contains zero a point $x$ is interior iff the value of Minkowski functional in that point is strictly less than $1$. is there anyone to help me.
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### When is a sequentially closed cone, closed?

Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What are criteria, that would imply that $C$ is closed (in the topology ...
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### Sets whose intersection with line segments have finite components

Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
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### Convex functions and uniform convergence of derivatives

Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions. Consider the sequence of derivatives $(f_n')_{n\in\mathbb{N}}$. Suppose that ...
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### affine set convex set

How to show the following: Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if $\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
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### About the problem 20 chap 3 (functional analysis, Walter Rudin) [duplicate]

Possible Duplicate: I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed. Let $\{u_1,u_2,u_3,\dots \}$ be sequence of pairwise orthogonal unit vectors in Hilbert ...
This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima. "Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...