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Math Help - Find the volume of this region!

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    Find the volume of this region!

    Suppose we are working in \mathbb{R}^n and that we write x \in \mathbb{R}^n in the form x=(x_1,...,x_n). Find the volume of the region

    0 < x_1 < x_2 <...<x_j<1
    x_{k}<1 \: \: \mbox{for } j<k\leq n

    for any 1 \leq j \leq n.
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  2. #2
    Moo
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    A Cute Angle Moo's Avatar
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    Hello,
    Spoiler:
    I think it's called the simplex.
    The volume for 0 < x_1 < x_2 <...<s ~,~ s>0 is \frac{s^n}{n!}

    So here it's \frac{1}{n!}



    You can think of it as the volume of [0,s]^n, and that it's just a matter of permuting the coordinates...
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  3. #3
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    Moo is correct. That volume can be written as the repeated integral
    \int_{x_n= 0}^1\int_{x_{n-1}= 0}^{x_n}\int_{x_{n-2}= 0}^{x_{n-1}+ x_n}\cdot\cdot\cdot\int_{x_1= 0}^{x_2+ x_3+\cdot\cdot\cdot x_n} dx_1dx_2\cdot\cdot\cdot dx_{n-1}dx_n
    and Moo's answer can be proved by induction on n.
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  4. #4
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    I don't think it's the simplex, since, for example, in \mathbb{R} ^2 if j=1 then the region R= \{(x_1,x_2) \in \mathbb{R} ^2 : 0<x_1 <1, x_2 <1 \} is unbounded (actually it is an open strip cut at x_2=1) and it's area is not finite (Acutally I believe it's only finite if j=n).
    On another note, if I recall correctly a simplex is a set of the form S(a_1,...,a_n)= \{ x=(x_1,...,x_n) \in \mathbb{R} ^n : x= \sum_{i=1}^n \ {b_i a_i} where b_i \in [0,1] and 1= \sum_{i=1}^n \ b_i \} where a_i \in \mathbb{R} ^n
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