Abstract This paper investigates Markov-invariant geometric structures on the denormalized state space, the space of strictly positive measures on a finite set. Within the subclass of structures compatible with the classical Nagaoka–Amari structure, we characterize all flat structures on the denormalized state space and identify a distinct one that complements the classical Nagaoka–Amari structure. This companion flat structure sheds new light on the = 1 α = ± 1 symmetry of the probability simplex and is conformally related to the Nagaoka–Amari structure, providing a coherent framework for understanding their relationship.
Yoshitaka Fujiwara (Thu,) studied this question.