A counterexample to Matsumoto's conjecture regarding absolute length vs. relative length in Finsler manifolds

Abstract

Matsumoto conjectured that for any Finsler manifold $(M, F)$ for which the restriction of the fundamental tensor to the indicatrix of $F$ is positive definite, the absolute length $F(X)$ of any tangent vector $X \in T_xM$ is the global minimum for the relative length $|X|_y$ as $y$ varies along the indicatrix $I_x \subset T_xM$ of $F$. In this note, we disprove this conjecture by presenting a counterexample.