# 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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### Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \frac{1}{\mu(E)} \int_{E} f d \mu \end{...
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### On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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### Supporting hyperplane to a compact, convex set in Hilbert space at a given boundary point

Does one always exist? I see that it is true in finite dimensions.
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### Permutation and Combinatorics Problem

A function $G$ is defined on a set $S$ with size $k$ : $G(a_1,a_2,a_3,\ldots,a_k)$. $G(a_1,a_2,a_3,\ldots,a_k) = 1$ if and only if a convex polygon can be created by taking these $k$ elements as the ...
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### Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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### Largest and smallest shape enclosed within circles

There is a convex shape $C$. It is known that the largest disc contained in $C$ has radius $r$ and the smallest disc containing $C$ has radius $R$. What are the smallest-area and largest-area shapes ...
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### Characterization of concave function on $\mathbb{R}^N$

I need help with the following problem, I have no idea how to proceed: Let $u \colon \Omega \subseteq \mathbb{R}^N \to \mathbb{R}$ a continuous function, where $\Omega$ is open, connected and bounded....
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### Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
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### mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpinski's theorem from which we can deduce that for ...
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### Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
A piece-wise constant or blocky signal can be defined as follows Definition: Let $p,b\in\mathbb{N}$ such that $b\leq \left(p-1\right)$. Define the set of normalized blocky vectors as the following \...