Let $X$ be a random variable with a symmetric support $S\subset[-M,M]$ for some $M>0$. (i.e., if x is a point of increase of CDF $F_X(\cdot)$, so is $-x$.)

Has the infimum value of $c$ such that
\begin{equation} 
X\leq_1 -X+c
\end{equation}
been discussed in literature? Any non-trivial upper/lower bounds on $c$? 

($A\leq_1B$ means that $B$ first order dominates $A$.)