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Fix $n$ and consider the polytope given by the inequalities $$x_i\leq x_j, \text{ and } 0 \leq x_i \leq a_i \text{ for all } 1\leq i<j \leq n,$$ where $a_i \leq a_i\leq \dots \leq a_n$ are fixed positive integers.

Has this family of polytopes been studied anywhere? Do they have a name? Is there a general formula for its (normalized) volume?

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  • $\begingroup$ @Richard Stanley: That is nice! Is there a known hook-like formula for determining its volume? The above polytope has a unimodular triangulation. Also, adding $b_i \leq x_i$ as an extra condition, has this also been considered somewhere. (This can be seen as a face of some Gelfand-Tsetlin polytope). $\endgroup$ – Per Alexandersson Dec 11 '14 at 22:05
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    $\begingroup$ There is no simple hook-like formula for the volume. See Theorem 1 and Theorem 11 of the paper cited in my answer. $\endgroup$ – Richard Stanley Dec 12 '14 at 0:54
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Set $x_i=y_1+\cdots+y_i$ and $a_i=x_1+\cdots+x_i$. We then we get the so-called "Pitman-Stanley polytope" (http://front.math.ucdavis.edu/9908.5029).

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Just to see what it looks like for $n=3$, here I used $(a_1,a_2,a_3)=(1,2,3)$:


          PolytopeIneq


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