$\def\R{\mathbb{R}}$ $\def\dd{\mathrm{d}}$ I wish to prove the following: Let $X$ be a connected set of $\R^n$ and $f: X \to [0, \infty)$ a continuous function with a unique global maximum point $t^*$ and a finite set of local maxima. Let $$ t_k = \frac{\int_X t[f(t)]^k\, \dd t}{\int_X [f(t)]^k\, \dd t}. $$ Then $$ t_k \to t^* \mbox{ as } k \to \infty. $$ I'm also okay with a proof taking $X$ as a bounded interval of $\R$.
The idea with $t_k$ is to have something close to softargmax, but for non-discrete domains. To prove the statement, I thought of trying to show the integrand converges to another sequence that we know to converge to a Dirac delta, but I've failed to get anywhere with this idea.
Any help is very appreciated.