# 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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### 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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### An inequality involving expectation and a concave function

I have a real-valued random variable $X$ and a function defined on the real line $x \mapsto U(x)$ which is concave and strictly increasing. What I am wondering is whether the following is true. For ...
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### When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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### Concave up theorem for $f:A \to\mathbb R, A \subseteq \mathbb R$ - True or false?

I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a ...
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### Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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### Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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### Are odd functions that are concave and increasing everywhere necessarily linear?

The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions. I think that if an ...
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### What value of 'a' will be the function is convex, concave or not either?

$$f(x,y) = -6x^2 + (2a+4)xy - y^2 + 4ay$$ The solution has to be : $$-2-\text{gyök}(6) \leq a \leq -2 + \text{gyök}(6)$$ I tried to define the derivation of the function accordance with $x$ , and ...
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### Caratheodory's theorem for a point in boundary

I am wondering whether the following holds: if $x$ in $\mathbb{R}^d$ lies in the boundary of the convex hull of a set $P$, then $x$ can be expressed as a convex combination of $d$ points in $P$. We ...
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### How to get inequality representation of cone give by generators

To be as specific as possible, I have a subspace $I \subset R^{14}$ of dimension $3$ and an $11\times14$ matrix $A$ with linearly independent rows spanning the orthogonal complement of $I$. I have an ...
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### Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$\operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}.$$ (it is a cone in ...
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### Dual of a point which is in the convex cone of a set, contains the dual cone of that set

Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...