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There don't seem to be a lot of high-dimensional shapes whose volume, surface area, etc. can be expressed in a concise way. The examples I know of are:

  1. Spheres
  2. Cubes (or parallelotopes, more generally)
  3. Simplices
  4. Zonotopes

What other classes of high dimensional objects admit relatively simple volume (or area, etc.) formulas?

EDIT: Since zonotopes are the most unfamiliar of my examples, here's a reference: Chapter 9 of "Computing the Continuous Discretely". To summarize, a zonotope is a set of the form $$\{a_1 \vec{x}_1 + \cdots + a_m \vec{x}_m \:|\: a_1,\dots,a_m\in[0,1]\}$$ where $\vec{x}_1,\dots,\vec{x}_m\in \mathbb{R}^n$ are fixed. This is like a parallelotope, except the vectors $\vec{x}_j$ need not be linearly independent (e.g. $m$ can be greater than $n$). The volume of such a zonotope is given by $$ \sum_{S\subset \{1,\dots,m\}, |S|=n} |\det[x_i]_{i\in S}|$$ which means: "Take any n of the m vectors $\vec{x_i}$ and compute the volume of the parallelotope formed by these n vectors in $\mathbb{R}^n$. Sum over all such parallelotopes and you get the volume of the zonotope."

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    Can you share a formula for the volume of zonotopes? – 2017-02-21

2 Answers 2

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(1) A minor addition to your list: Ellipsoids are just linear transformations of spheres. If $M$ is a linear transformation applied to sphere $S$, then the volume of the ellipsoid is vol$(S) \cdot$ det$(M)$.

Here is an explicit reference: Wilson, A. John. "Volume of n-dimensional ellipsoid." Sciencia Acta Xaveriana. 2009.

EllVol
That must be $a_1 a_2 a_3 \cdots$. Typo: commas → multiplication.

(2) I found this reference but not the paper itself:

"The $(n+1)$-Volume of an $n$-Torus." Link.

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A couple more examples I've found:

1) Cross-polytopes. These are generalizations of an octohedron. Wikipedia has a nice article on them, and the standard n-dimensional cross-polytope has volume $\frac{2^n}{n!}$.

2) Cones. A cone is formed from a base shape of codimension 1 and a point at some height $h$ above the base. In n dimensions, if the base has $(n-1)$ volume $A$, then the cone has $n$ volume $\frac{Ah}{n}$.