# Tagged Questions

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### Proving convexity

I ask you please some help with this problem: Let A $\subseteq$ R$^n$ be a convex set and $C(A)$ = {$\lambda$x, $\lambda \in \mathbb{R}$, $\lambda \geq 0$, x $\in$ A}. Prove that $C(A)$ is convex. ...
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### How to prove that is a cone

I have to prove that the set $$K=\lbrace u\in C[0,1]\mid u(t)\geq a(t)||u|| \text{ on } [0,1]\rbrace,$$ where $a(t)=\displaystyle\frac{t(2p-t)}{p^2}$ with $\frac12<p<1$, and ...
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### Why is this composition of concave and convex functions concave?

Please forgive my ignorance. I have a quick silly question about a statement given without proof in Convex Optimization by Boyd and Vandenberghe (page 87). Suppose $\mathbb{R}_+^n$ is the set of ...
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### Is the set of all concave functions a convex set?

How can I prove this? I saw a similar question here: (But this was only for when g(x) is ≥0) Prove that a set defined by concave functions on $R^n$ is convex
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### proving compactness and convexity of a set

Suppose functions $f(x)$ and $g(x)$ are continuous with domain $X \subset \mathbb{R}$ which is nonempty, convex and compact, can we show that $$S \equiv (f(x), g(x))$$ for all $x \in X$ is ...
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### Is $C(B^c)$ an open set?

Assume that $B$ is an open set, if $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1,x_i \in B\}$ is a convex that contains $B$, $C$ is an open set? What's ...
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### Prove that $\text{int}(\text{dom}(f))$ is a convex set.

Let $f$ be a convex function. I have to prove that $\text{int}(\text{dom}(f))$ is a convex set. (Be careful with $-∞$ )
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### fail of cantor intersection property on closed , bounded , convex sets of integrable functions

This is from my recent homework. I am asked to find a descending nested sequence of closed , bounded , nonempty convex sets $\{D_n\}$ in $L^1(\mathbb{R})$ such that the intersection is empty , where ...
Here is part of the question in my HW. Let$\ \{C_n\}\subset X$ be a bounded nested decreasing sequence of closed and convex sets. I am asked to show that $\bigcap C_n \not= \emptyset$ iff X is ...
Can anyone help me with this problem? Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form ...