Abstract: The linear interpolation convexity (LIC) has served as the essential condition to guarantee the uniqueness of minimizer for pairwise interaction energies. In particular, for power-law potentials $W(x) = frac{|x|^a}{a} - frac{|x|^b}{b}$, it is known that LIC holds for $-d lt b leq 2, 2 leq a leq 4, b lt a, (a, b) neq (4, 2)$. We extend the notion of LIC by requiring the energy convexity only for linear interpolation between probability measures supported on a prescribed ball. This allows us to prove the uniqueness of minimizer for power-law potentials with $a$ slightly smaller than 2 or larger than 4.