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 bounds?

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

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 $A$.)