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In this paper, we consider probability measures and on a -dimensional sphere in ${𝐑}^{d+1},d\ge 1,$ and cost functions of the form $c(𝐱,𝐲)=l\left(\frac{{|𝐱-𝐲|}^2}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if and vanish on $(d-1)$-rectifiable sets, if ${lim}_{t\to 0^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left(l^{\text{'}}\left(t\right)\right)}^2$ is monotone then there exists a unique optimal map that transports onto $\nu ,$ where optimality is measured against Furthermore, ${inf}_{𝐱}|T_o𝐱-𝐱|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ existence of optimal...