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In the area of convex analysis and the area of optimization in their general sense, are convex subsets assumed to be in vector spaces or topological vector spaces? Are convex functions defined to be from convex subsets in vector spaces or topological vector spaces to general field or some special field such as $\mathbb{R}$ or $\mathbb{C}$?

Now instead of considering the general sense, what are the special cases that have been extensively studied and have significance in application?

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Q1: Are convex subsets assumed to be in vector spaces or topological vector spaces?

Convex sets are easily defined in topological vector spaces over a subfield of $\mathbb{C}$, following the usual definition: $A$ is a convex set if $\forall x,y\in A$ we have $tx+(1-t)y\in A$ for $t\in[0,1]$. Basically means we can draw a line between any two points and the line segment is contained in $A$.

For non-euclidean spaces convexity generalises to geodesically convex set see "Convex functions and optimization methods on Riemannian manifolds" by Constantin. There are a few other generalisations. For the most abstract there are Convexity spaces which is an axiomatic description but I can't remember much about this.

Q2: Are convex functions defined to be from convex subsets in vector spaces or topological vector spaces to general field or some special field such as R or C?

Firstly I have seen a convex function defined without the need for its domain to be a convex set at all! It required the function to only be convex on any convex subset of its domain. (I don't think is standard though.) Secondly, let $f:I\rightarrow K$ be a convex function where $I$ is a convex subset.

If $K=\mathbb{R}$ then $f$ is convex iff $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$, $t\in[0,1]$. Is suspect this could be generalised to any ordered field.

If $K=\mathbb{C}$ then we can't use this definition as $\mathbb{C}$ is not a totally ordered field. Instead we have $\mathbb{C}\sim\mathbb{R}^2$ and use the hessian: $f$ is convex iff $\nabla^2f$ is positive semi-definite. However this requires $f\in\mathcal{C}^2$.

Q3: What are the special cases that have been extensively studied and have significance in application?

The majority of the applications in convex optimisation are with a real vector space and a convex real-valued function. There are so many applications for this the mind boggles. A short memory dump is:

  • Least squares problems
  • Linear programming: network flow problems, mircoeconomics.
  • Quadratic programming: This forms the basis of so many optimization algorithms
  • Entropy maximisation
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Regarding Q1: You don't need any topology to define or discuss convexity. A vector space setting is sufficient. Vector spaces over the real field are the most reasonable, by far. If $t$ is a complex number, then saying that something is true for all $t \in [0,1]$ is a bit odd. If $t$ belongs to some other field (neither the real nor the complex field), then it's not at all clear what $t \in [0,1]$ means. –  bubba Jan 6 '13 at 3:05
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