Extended convexity and uniqueness of minimizers for interaction energies

Linear interpolation convexity (LIC) has served as the crucial condition for the uniqueness of interaction energy minimizers. We introduce the concept of the LIC radius which extends the LIC condition. Uniqueness of minimizer up to translation can still be guaranteed if the LIC radius is larger than the possible support size of any minimizer. Using this approach, we obtain uniqueness of minimizer for power-law potentials $W_{a,b}({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b},\,-d<b<2$ with $a$ slightly smaller than 2 or slightly larger than 4. The estimate of LIC radius for $a$ slightly smaller than 2 is done via a Poincar\'{e}-type inequality for signed measures. To handle the case where $a$ slightly larger than 4, we truncate the attractive part of the potential at large radius and prove that the resulting potential has positive Fourier transform. We also propose to study the logarithmic power-law potential $W_{b,\ln}({\bf x}) = \frac{|{\bf x}|^b}{b}\ln|{\bf x}|$. We prove its LIC property for $b=2$ and give the explicit formula for minimizer. We also prove the uniqueness of minimizer for $b$ slightly less than 2 by estimating its LIC radius.