Rounding of Polytopes in the Real Number Model of Computation

Let A be a set of m points in Rn. We show that the problem of 1 + en-rounding of A, i.e., the problem of computing an ellipsoid E ⊆ Rn such that [1 + en]-1E ⊆ conv. hullA ⊆ E, can be solved in Omn2e-1 + ln n + ln ln m arithmetic operations and comparisons. This result implies that the problem of approximating the minimum volume ellipsoid circumscribed about A can be solved in Om3.5 lnme-1 operations to a relative accuracy of e in the volume. The latter bound also applies to the 1 + en-rounding problem. Our bounds hold for the real number model of computation.