Equivalence of weak and strong modes of measures on topological vector spaces

Modes of a probability measure on an infinite-dimensional Banach space $X$ are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of $X$, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces.