# Tagged Questions

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### Looking for an entry level discussion on convex analysis

I have been studying for a qualifier and every so often I come across questions such as: Let $f_n:[a,b] \to \mathbb{R}$ be convex functions and suppose that $f(x) = \lim_{n \to \infty} f_n(x)$ exists ...
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### Can we say a convex cone is a closed set without further proof?

There are some related problems: 1. dual cone is closed 2. Why is any subspace a convex cone? Consider a cone $\mathcal{C}(A)$: $$\mathcal{C}(A) = \{Ax: x\geq 0\}$$ This is a cone generated ...
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### Proof of the Line Segment Principle for Convex Sets

I'm self studying a chapter on convex sets in preparation for a course on optimisation for economists however I'm having trouble understanding the proof of the line segment principle. I would ...
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### How to prove coercivity

I have a problem in understanding how to prove if a function is positive or negative coercive. I understood the definition of coercivity, which is: $$\lim_{||x|| \to +\infty}f(x) = +\infty$$ However, ...
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### Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
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### How to Compute or Find an Upper Bound for the Diameter of a Convex Set?

Let $\mathcal{R}$ be an $n$-dimensional bounded hyperrectangle, and consider a $n\times n$ matrix $A$ with real entries. Given set $\mathcal{R}$ and matrix $A$, I want to compute or find an ...
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### Is function $f(x) = \left( ( {\bf w} -x {\bf v})^T ( {\bf w} -x {\bf v}) \right)^k$ convex for $k \ge 1/2$
Let ${\bf v}$ and ${\bf w}$ be column vector of dimension $n$. Is function $f(x) = \left( ( {\bf w} -x {\bf v})^T ( {\bf w} -x {\bf v}) \right)^k$ convex for $k \ge 1/2$ ? I want to show this via ...