= Admissible values for a sandwich extension
{title2=$\sup_{x,y}[S(y)-p(x+y-z)+q(x)]\leq U(z)\leq\inf_{x,y}[-S(y)+p(x+y+z)-q(x)]$}
Suppose $S$ satisfies the mixed domination on a subspace $Y$, and $z\notin Y$. Apply that domination to $y+y'$ and $x+x'$, split the $p$ argument as $(x+y+z)+(x'+y'-z)$, and use superadditivity of $q$. This proves every lower candidate in the displayed interval is at most every upper candidate. The zero-vector candidates bound the lower supremum below by $-p(-z)$ and the upper infimum above by $p(z)$, so both endpoints are finite. Choosing any intervening value $a$ permits $U(y+tz)=S(y)+ta$. For $t>0$, scale the upper comparison by $t$; for $t<0$, scale the lower comparison by $-t$. Both give the required mixed domination, proving the interval characterizes the permitted extension values.
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