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I have a question regarding homeomorphisms from compact convex spaces to the standard simplex. I know that every compact convex subset of $\mathbb{R}^{n}$ with a non-empty interior is homeomorphic to the standard n-simplex $\Delta^n$, but I'm wondering if there is a closed-form formula you can write for this homeomorphism. Specifically, if $X$ is a compact convex subset of euclidean space with a non-empty interior and $\Delta^n$ is the standard simplex homeomorphic to this space, that means there is a continuous bijective function $f: X \to \Delta^n$. I want to know if there is a way you can "spell out" $f$ and not just leave it as an unspecified $f$.

I found some related questions that show such formulas, for example, this one from the n-simplex to the n-ball: How to show the standard $n$-simplex is homeomorphic to the $n$-ball or this one from the open n-ball to $\mathbb{R}^n$: Is an open $n$-ball homeomorphic to $\mathbb{R}^{n}$?

This gives me the intuition that such a closed-form exists, but I'm not able to figure it out so far. This area is not really my strong suit, so if there are any resources that you would suggest me to look into I would be very happy to study this further!

Willem Röpke
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  • Do you agree with the following definition of simplex? – Antonio Maria Di Mauro Jan 03 '22 at 13:27
  • Definition If $x_0,...,x_k$ are $(k+1)$ affinely indipendent points of $\Bbb R^n$ (which means that the vectors $(x_1-x_0),...,(x_k-x_0)$ are linearly independent) then simplex determined by them is the set $$ \mathcal S:=\Biggl{x\in\Bbb R^n: x=x_0+\alpha^i\vec v_i,,,\text{and},,, \sum_{i=1}^k\alpha^i\le1,,,\text{and},,,\alpha^i\ge0,,,\text{for all},i=1,\dots,k\Biggl} $$ where $\vec v_i:=(x_i-x_0)$ for each $i=1,\dots,k$. – Antonio Maria Di Mauro Jan 03 '22 at 13:27
  • The only thing I would say I have different in my definition of the standard simplex is that the sum of all coefficients should be equal to 1, not smaller than or equal to. – Willem Röpke Jan 03 '22 at 14:54
  • Okay, so with my definition it is possible to prove that $\mathcal S$ is a $k$-manifold with corners and its boundary is exactly your $\mathcal S$ so that we can conclude that your $\mathcal S$ is homeomorphic to the boundary of a $k$-cube and so of a $k$-sphere. – Antonio Maria Di Mauro Jan 03 '22 at 15:00
  • Moreover it is possible to prove that $\mathcal S$ is homeomorphic to an open set of a $k$-cube with only two map. – Antonio Maria Di Mauro Jan 03 '22 at 15:02
  • I do not know if this interest you but if you like you can find what I claimed here and moreover here it is also proved that any simplex not only a manifold with corners but it is also an orientable manifold with corners. – Antonio Maria Di Mauro Jan 03 '22 at 15:04
  • Thank you for the pointers I will check them out, but I think these results are too complex for what I'm actually trying to do. I'm more trying to see if there is some general formula for the homeomorphism rather than a proof of existence. – Willem Röpke Jan 03 '22 at 15:10
  • So your are searching a global homeomorphism from $\mathcal S$ to $\Bbb S^k$, right? – Antonio Maria Di Mauro Jan 03 '22 at 15:11
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    I'm not sure what your $\mathbb{S}^k$ is, but if $S$ is your simplex, then I'm trying to see if there is a formula to transform any compact convex subset of euclidean space to $S$ (so a homeomorphism) – Willem Röpke Jan 03 '22 at 15:15
  • Okay, only I suggest to say more explicitely in the question this: indeed it do not seems clear, at least to me. Anyway if I will some ideas I will tell you. – Antonio Maria Di Mauro Jan 03 '22 at 15:19
  • I just want to clarify that I'm really new to (algebraic) topology as my work is mostly in other areas. I believe there are some important links however and I'm trying to see if I can make these links concrete by writing down an explicit formula for the homeomorphism. – Willem Röpke Jan 03 '22 at 15:19
  • Probably I think you have to use something as the Brouwer fixed theorem or rather the Invariance of domain: unfortunately about Algebraic Topology for now I only studied something about homotopies to study the exact differential form on contractible sets so I can't help you immediately. – Antonio Maria Di Mauro Jan 03 '22 at 15:21
  • The question is how your convex subset is given. If it is given in the form ${x: p(x)\le 1}$ for a suitable function $p$ (that is regarded as "known"), then yes, see my answer here. – Moishe Kohan Jan 04 '22 at 00:36
  • Yes I saw that answer! As far as I understood it however, the formula in your answer is a homeomorphism to a ball and not to a simplex. – Willem Röpke Jan 04 '22 at 07:53
  • @WillemRöpke: Yes, but do you know how to compose maps? – Moishe Kohan Jan 05 '22 at 03:55
  • Ah so I can actually use your formula composed with the one I mentioned in the question and I have the homeomorphism? That would be a nice solution indeed! – Willem Röpke Jan 05 '22 at 07:06
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    It is only true compact convex subsets with non-emoyt interior (for example, one-point subsets are compact and convex). Moreover you have to replace $\mathbb R^{n+1}$ by $\mathbb R^n$. And in which sense do you expect that $f$ can be "spelled out"? If you work with an abstract $X$, what should an explicit formula be? – Paul Frost Jan 13 '22 at 14:40
  • Yes, you are completely right on those remarks and I will update them in the question! I want $f$ to be "spelled out" in terms of an actual formula that defines the projection for each point $x$, such as for example here: https://math.stackexchange.com/questions/538262/how-to-show-the-standard-n-simplex-is-homeomorphic-to-the-n-ball – Willem Röpke Jan 14 '22 at 15:13
  • In this first link, there is a relatively explicit construction of the homeomorphism in terms of rescalings along line segments. I'm not sure how much more f can be ``spelled out" in general, especially if X is not given in any explicit way (e.g., a collection of inequalities). – Simon Segert Jan 14 '22 at 15:53
  • Well, every link in the question and comments so far either shows the homeomorphism from compact convex sets to an n-ball or from an n-ball to a simplex. My question was specifically from a convex set to an n-simplex. I could also try to compose these maps, such as suggested above. – Willem Röpke Jan 16 '22 at 15:01

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