48 | | Once `affine_hull` completely respects the backend, this method will be used to calculate the induced volume by default (automatically). |

| 48 | {{{ |

| 49 | sage: P = polytopes.permutahedron(6, backend='normaliz') |

| 50 | sage: %time P.volume(measure='induced') |

| 51 | CPU times: user 37.7 s, sys: 42.4 ms, total: 37.7 s |

| 52 | Wall time: 24.2 s |

| 53 | 1296*sqrt(6) |

| 54 | sage: %time P.volume(engine='internal', measure='induced') |

| 55 | CPU times: user 42.9 s, sys: 71.9 ms, total: 43 s |

| 56 | Wall time: 41.8 s |

| 57 | 1296*sqrt(6) |

| 58 | }}} |

| 59 | |

| 60 | However, note that calculation of the affine hull takes quite some time in this case. So we leave the inexact normaliz option for now: |

| 61 | {{{ |

| 62 | sage: %time P.volume(engine='normaliz', measure='induced') |

| 63 | CPU times: user 710 ms, sys: 0 ns, total: 710 ms |

| 64 | Wall time: 92.3 ms |

| 65 | 3174.5387066469984 |

| 66 | }}} |

| 67 | |

| 68 | Possible follow ups: |

| 69 | |

| 70 | There is probably a good and fast way to convert the lattice volume to the euclidean volume exactly. Also, once we can set up a normaliz polyhedron with both Vrep and Hrep, affine hull should be really quick. |