  In the recent years, the notion of mixability has been developed with
applications to optimal transportation, quantitative finance and operations
research. An $n$-tuple of distributions is said to be jointly mixable if there
exist $n$ random variables following these distributions and adding up to a
constant, called center, with probability one. When the $n$ distributions are
identical, we speak of complete mixability. If each distribution has finite
mean, the center is obviously the sum of the means. In this paper, we
investigate the set of centers of completely and jointly mixable distributions
not having a finite mean. In addition to several results, we show the (possibly
counterintuitive) fact that, for each $n \geq 2$, there exist $n$ standard
Cauchy random variables adding up to a constant $C$ if and only if
$$|C|\le\frac{n\,\log (n-1)}{\pi}.$$
