# 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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### Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
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### About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...
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### Can a non-convex set be partitioned into a set of nearly convex subsets? [closed]

Consider a non-convex bounded subset $S \subseteq \mathbb{R}^{n}$. Is it always possible to partition this set into a finite set of disjoint subsets S = \bigcup_{i=1}^{n}s_i, \quad ...
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### Is a distance function on $\mathbb{R}^n$ convex?

Fix $z \in \mathbb{R}^n$. Let $||\cdot||$ be a norm on $\mathbb{R}^n$, and define the distance function $f(x)=||z-x||$ for $x\in \mathbb{R}^n$ Then, is it true that $f(x)$ is convex?
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### Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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### Show that f(x) is convex

Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions. Can I just go about this by using the regular definition of a convex function or which ...
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### Convexity/concavity of a strictly increasing and continuous function

Consider a continuous, strictly increasing function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ with $f(0)=0$, and $x>f(x)$ for all $x>0$. Is this enough to conclude anything about convexity/...
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### Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
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### compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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### Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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### Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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