Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let
$$D(n)=\{(d\_{1},\dots,d\_{n})\in\mathbb{Z}\_{+}^{n}:d\_{1}\geq\dots\geq d\_{n},\ \sum\_{i=1}^{n}d\_{i}\ \text{is even},\ \sum\_{i=1}^{k}d\_{i}\leq k(k-1)+\sum\_{i=k+1}^{n}\min\{k,d\_{i}\}\ \text{for all}\ 1\leq k\leq n \}.$$
One can observe that the diameter of $D(n)$ equals $(n-1)\sqrt[]{n}$. The question is:



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> How to compute the volume of the convex hull of $D(n)$?
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Volume of the convex hull of the set of all graphic sequences of a given length
The polytope considered by Richard Stanley is the polytope of degree sequences.
Sergiy Kozerenko's question is about the polytope of degree partitions for which see paper R46 of Volume 13 (2006) of the Electronic Journal of combinatorics. It is shown there that the volume of the polytope of degree sequences is n! times the volume of the polytope of degree partitions.


