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answered Triangulation with simplices of same volume
1d
comment Triangulation with simplices of same volume
I don't think the question is about triangulation by geodesic simplices. Otherwise, there would be a simpler answer as soon as $n\ge3$: most manifolds don't have totally geodesic submanifolds, so we don't have any geodesic simplex.
Apr
26
comment What determinant application can I show to a 16 year old?
I'm voting to close this question as off-topic because it should be asked on matheducators.stackexchange.com
Apr
26
comment Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem
This assumes the axiom of choice, which should be mentioned given the dependency of related problems to the chosen axioms.
Apr
26
comment Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem
The graph $G$ is connected (e.g. any two points can be connected by a chain of length $2$ if they are at distance $<1$). Despite the accepted answer, your question still makes sense in axiomatic theories without choice.
Apr
22
comment Has there been a computer search for a 5-chromatic unit distance graph?
An interesting variation, which seems more amenable to computer search, is to look for non-4-colorable almost-unit distance graph. You can indeed ask for the chromatic number $\chi_\epsilon$ of the graph whose vertices are the points of the plane, and an edge joins two vertices if their distance is between $1$ and $1+\varepsilon$. Obviously $\chi_\varepsilon$ decreases when $\varepsilon\to 0$, but the limit is not known. It is not known if the limit is the chromatic number of the plane either.
Apr
21
revised Compact manifolds locally bi-Lipschitz to Euclidean space
Added an alternative, more powerful argument.
Apr
21
comment Compact manifolds locally bi-Lipschitz to Euclidean space
@DeaneYang: I think you can avoid the exponential and deal with a continuous metric by controlling the length of a curve that goes out of $D_i$: replace $\frac12B_i$ by two balls, one small and the other large; define $k$ and $K$ using the latter, and say that the $g$ geodesic connecting two points of the image of the small ball either stays in the image of the large one (in which case we are done), or must be quite long to go out and back; choosing (a posterori) the small ball small enough, you can ensure that this can never happen. This may need some care on the order of quantifiers.
Apr
21
answered Compact manifolds locally bi-Lipschitz to Euclidean space
Apr
20
comment Survivor sets for expanding maps of the interval
Also, I have a problem with variables: there are a $n$ outside the set you are defining, and a quantified $n$ inside it, so I guess there should be two different numbers and it is not clear what you mean.
Apr
20
comment Survivor sets for expanding maps of the interval
I am not sure that this matters much, but I guess you mean either "piecewise smooth" or you want to replace the interval with the circle (otherwise, there is no such map).
Mar
27
awarded  Yearling
Mar
22
comment Invariant probability on a unit ball of a Banach space
A sufficient condition is that $\Gamma$ preserves a finite-dimensional subspace. If this is too trivial for your taste, consider replacing "non-atomic" by "not supported on a finite-dimensional subspace".
Mar
17
comment Extending continuous functions from $\partial X$ to $X\cup \partial X$
Yes, I think that is what I had in mind (I have not worked with this myself, I just remembered that Besson-Courtois-Gallot have used these measure to prove their celebrated rigidity result). The reference Pacific J. Math. Volume 159, Number 2 (1993), 241-270 by Coornaert seems relevant by I have trouble retrieving it to check.
Mar
16
comment Extending continuous functions from $\partial X$ to $X\cup \partial X$
A possible strategy would be to associate to each point $x\in X$ a probability measure $\mu_x$ on $\partial X$, such that the measure depends continuously on the point and converges to $\delta_\zeta$ when $x$ converges to $\zeta\in\partial X$. I think such construction exist at least in particular cases (e.g. limit of the uniform measure on spheres, or using the critical exponent maybe). Then you extend $f$ by $\int f d\mu_x$.
Mar
9
comment Robustly recurrent random walk
Since the answers below are to your original question, I took the liberty to mention the part that I understood was added. Fix if I was wrong, and note that it is better to make edits more explicit (I had much trouble understanding the answers after reading the edited version of the question).
Mar
9
revised Robustly recurrent random walk
Precised the edit to make answers understandable
Mar
7
comment Majority coloring for directed graphs
Nice concept, but you should make clear what your question is.
Mar
5
comment Ordinal of injectivity for a smooth regular curve with a finite arc-length
The question is still missing something. The sup could be on the empty set, I doubt the closed intervals are really what is meant, there is no obvious link between the formal definition and injectivity, and the last coordinates presently only have as a consequence to make the regularity assumption vacuous. After that many edits and tentative to make sense out of it, I think already too much time have been spent on this by others than the asker. I vote to close.