  This paper analyzes the convergence rates of the {\it Frank-Wolfe } method
for solving convex constrained multiobjective optimization. We establish
improved convergence rates under different assumptions on the objective
function, the feasible set, and the localization of the limit point of the
sequence generated by the method. In terms of the objective function values, we
firstly show that if the objective function is strongly convex and the limit
point of the sequence generated by the method lies in the relative interior of
the feasible set, then the algorithm achieves a linear convergence rate. Next,
we focus on a special class of problems where the feasible constraint set is
$(\alpha,q)$-uniformly convex for some $\alpha >0$ and $q \geq 2$, including,
in particular, \(\ell_p\)-balls for all $p>1$. In this context, we prove that
the method attains: (i) a rate of $\mathcal{O}(1/k^\frac{q}{q-1})$ when the
objective function is strongly convex; and (ii) a linear rate (if $q=2$) or a
rate of $\mathcal{O}(1/k^{\frac{q}{q-2}})$ (if $q>2$) under an additional
assumption, which always holds if the feasible set does not contain an
unconstrained weak Pareto point. We also discuss enhanced convergence rates for
the algorithm in terms of an optimality measure. Finally, we provide some
simple examples to illustrate the convergence rates and the set of assumptions.
