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I was looking for a standard name of the set $\{x \in A : x_1 \leq x_2 \leq \cdots \leq x_n\}$, where $A = [0,1]^n$ or $A = [0,\infty)^n$. I think I saw this recently, but now I cannot find it anywhere.

Thanks in advance.

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    @Kannappan: No, I mean that each coordinate is restricted to be bigger than the previous one.2012-03-30
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    You could call it the set of all (weakly) increasing, or non-decreasing, $n$-tuples from $[0,1]$ or $[0,\infty)$.2012-03-30
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    @Kannappan: The $x_i$ are the components of $x$.2012-03-30
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    Thank you @joriki It clears up now. :-)2012-03-30
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    The best thing, I think, is **not** to introduce a name for such a thing (unless you are about to start writing a complete book about it...)2012-03-30
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    @MarianoSuárez-Alvarez: Probably true. My main reason for wanting a name is to do literature searches more easily.2012-03-30
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    If you want to search the literature it is better to explain what you want to look for... *what* do you want to know/do with this set? You have not even told us if you are looking at it from the representation theoretic point of view, or that of combinatorial optimization, or something else —and the relevant keywords and references will wildly vary with the point of view! «Weyl chamber truncated at level 1» will be of very little help —if any!— in searching for literature on integral linear optimization...2012-03-30
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    That's a good point. I am interested in solving a nonlinear optimization problem with the solution constrained to lie in the set in question.2012-03-30
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    In the case $A=[0,1]^n$ this is an $n$-simplex.2012-03-30

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The second one is a Weyl chamber, the first one a Weyl chamber (for the same root system) truncated at level 1.

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    Odds are, unless the context cries for this nomenclature, using it will not clarify anything —and I am pretty sure that if the term ringed a bell for the intended audience and/or the OP then the question would not have been asked :)2012-03-30
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    @Didier: Thanks! This is the name I couldn't remember. :)2012-03-30
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    @Mariano: Bingo! :-)2012-03-30
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    @SetMaster: You are welcome.2012-03-30
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In geometric terms this is an n-simplex called a Schläfli orthoscheme when $A = [0,1]^n$.

The case when $A = [0,\infty)^n$ is an unbounded convex region, but very similar to a simplex in that it is simply a limit of the simplex as coordinates of the first figure are scaled up. One may well refer to this figure as the non-negative cone of the respective $(n-1)$-simplex.

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    If someone started talking to me about an «$n$-simplex with an orthogonal corner» I would surely ask what she means...2012-03-30
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    I thought the [simplex](http://en.wikipedia.org/wiki/Simplex#The_standard_simplex) was defined by the *sum* of the coordinates being 1, not by the coordinates being nondecreasing.2012-03-30
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    @Didier: A regular n-simplex can be construed as a sum of n+1 coordinates that add up to 1. However this simplex is not regular in the sense of having all faces and angles congruent.2012-03-30
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    @MarianoSuárez-Alvarez: In my rush to improve, I went wrong! New term, new link.2012-03-30
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it is the weyl chamber for the symmetric group

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    Could you provide some more information?2012-04-03