  Kadeishvili's minimal model theorem establishes the existence of an
$A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg
associative algebra, which captures its homotopy type. In this note we prove
the existence of minimal models that are unique up to isotopy, a stronger
result obviously known to T. Kadeishvili and certainly to others, yet seemingly
overlooked by mankind. We will explore how this stronger result can help in the
study of Massey products. First, we show that the attempts to extract a local
information from the ternary operation $\mu_3$ of our minimal model leads
directly to the rediscovery of the triple Massey product. The motto is: "The
triple Massey product is an invariant manifestation of $\mu_3$." We then prove
that, under reasonable assumptions, the higher Massey product $\langle
x_1,\ldots,x_n\rangle$ equals the set of all values $\mu_n(x_1,\ldots,x_n)$,
where $\mu_n$ runs over the $n$-ary products of our minimal models. We believe
that this note will help to elucidate the still somewhat enigmatic relationship
between minimal models and Massey products.
