Intrinsic Hilbert metrics on cones of equivalent norms

Abstract

Let $X$ be a Banach space and let $\mathcal{N}(X)$ denote the family of equivalent norms on $X$. We study Hilbert projective metrics on its projectivization $\mathcal{N}'(X)$ induced by ambient cones of nonnegative functions, with particular attention to the intrinsic Hilbert metric induced by the cone $\mathcal{N}(X)\cup\{0\}$. First, we show that the symmetric logarithmic metric on $\mathcal{N}'(X)$ is the Hilbert projective metric induced by the cone of nonnegative real-valued functions, and we characterize the ambient cones that induce the same metric. We then study the intrinsic order on $\mathcal{N}(X)$. For this purpose, we introduce the triangular defect $Δ_p(x,y)=p(x)+p(y)-p(x+y)$ and prove that two norms $p,q\in\mathcal{N}(X)$ belong to the same intrinsic part if and only if their triangular defects are uniformly comparable, that is, $aΔ_p\leqΔ_q\leq bΔ_p$ for some $a,b>0$. This yields an explicit formula for the intrinsic Hilbert metric in terms of the pointwise comparison of the norms and of their triangular defects. Finally, the map $Φ(p)=(p,Δ_p)$ realizes the intrinsic cone order of $\mathcal{N}(X)$ inside a canonical product cone of nonnegative functions and preserves the Hilbert metric on each intrinsic part.