Total order on $\mathbb{R}^n$ such that $\{x \in \mathbb{R}^n \mid x < t\}$ is convex for all $t$ I'm currently working on stuff where I state some of my results in terms of the lexicographical minimum point of a compact convex set. I noticed that the only properties I need from my ordering $<$ (here lexicographical) are the following:


*

*The order is a total order.

*For all $t\in \mathbb{R}^n$, $\{x \in \mathbb{R}^n \mid x < t\}$ is convex.

*Every compact convex set has a smallest point


So I'm wondering what other "nice" order I might use, that has these properties. By "nice" I mean typically stuff that could be used in a computational context, not things constructed using the axiom of choice for example.
Right now the only examples I could come up with are the standard lexicographical order, or the lexicographical order in other coordinate systems. Can you think of any other? Examples for specific dimensions n>1 (say n=2 or 3) are also welcome.
 A: You can construct a lot of others by generalizing the reasoning you already have using pullbacks of orders. In particular, let $\preceq $ on $\mathbb R^n$ be the lexicographic order that you already have. If you have any set of convex functions $f_1,\ldots,f_n:\mathbb R^m\rightarrow \mathbb R$ such that the pairing map given by $f(v)=(f_1(v),\ldots,f_n(v))$ is injective, then defining
$$v_1\leq v_2 \Longleftrightarrow f(v_1) \preceq f(v_2)$$
is an example of such an order. Property (1) is trivial, property (2) follows from the convexity of $f_1,\ldots,f_n$ and property (3) follows from the fact that $f_1,\ldots,f_m$ are continuous (which follows from convexity).
This yields some orders that look pretty different; for instance, you could map $\mathbb R^2$ into $\mathbb R^3$ by the functions $(x,y)\mapsto (x^2+y^2,\,x,\,y)$ which gives an order that "spirals out" from the origin (and even has stronger properties like that all non-empty closed sets have a minimum)

It's also maybe worth noting that for any strictly convex norm $\|\cdot \|$, any order $\prec$ with the property that $x \prec y$ if $\|x\| < \|y\|$ satisfies your conditions just because all sets $S$ which contain an open ball and are contained in its boundary are convex and because all compact convex sets will have a unique minimum in the norm - hence in the order. This gives you a real lot of examples and shows that the previous conditions are stronger than necessary.
