There is a simple way to do so directly in Sage:

```
sage: p = polytopes.permutahedron(4); p
A 3-dimensional polyhedron in ZZ^4 defined as the convex hull of 24 vertices
```

Naturally, it is lower dimensional, so its volume is zero:

```
sage: p.volume()
0
```

But, changing the `measure`

to `'induced'`

, we can directly compute the Lebesgue measure inside of the affine hull, without doing any transformation:

```
sage: p.volume(measure='induced')
32
```

The `.volume`

method can use different algorithms and other measures (for example for lattice polytopes) whose performance may vary, you can check the accessible algorithms by typing `p.volume?`

and read the documentation for more details.