Volume formulas in L_p-spaces

We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments m 1/n 1/pn vol (∑ ⊕_p[-x_i, x_i]) ∼_cp n^-1/p′ (∑ | det(x_i)_i∈I|^p), i=1 card(I)=n where x₁, . . . , x_m are m vectors in ℝⁿ, 1/p + 1/p′ = 1. According to the definition of Firey, the Minkowski p-sum of segments is given by m m m 1/p′ ∑ ⊕_p[-x_i, x_i] = {∑ α_ix_i| (∑ |α_i|p^′) ≤ 1}. i=1 i=1 i=1