A lower bound on the barrier parameter of barriers for convex cones

Let K ⊂ R be a regular convex cone, let e1, . . . , en ∈ ∂K be linearly independent points on the boundary of a compact affine section of the cone, and let x∗ ∈ K be a point in the relative interior of this section. For k = 1, . . . , n, let lk be the line through the points ek and x ∗, let yk be the intersection point of lk with ∂K opposite to ek, and let zk be the intersection point of lk with the linear subspace spanned by all points el, l = 1, . . . , n except ek. We give a lower bound on the barrier parameter ν of logarithmically homogeneous self-concordant barriers F : K → R on K in terms of the projective cross-ratios qk = (ek, x ∗; yk, zk). Previously known lower bounds by Nesterov and Nemirovski can be obtained from our result as a special case. As an application, we construct an optimal barrier for the epigraph of the || · ||∞-norm in R n and compute lower bounds on the barrier parameter for the power cone and the epigraph of the || · ||p-norm in R .