Suppose we have a set of vectors on a unit circle $u_i$. And we use the Gaussian kernel to measure the distance among them, which could be described as:
$$ G(u,v)=e^{-||u-v||_2^2} $$
One already proved theory is that when we maximize the sum of distances among these vectors, the final result is the uniform distribution on the unit circle. This can be described as:
$$ U_n=\arg\min\limits_{u_1,u_2,...,u_N} \sum_{1\leq i\le j \leq N}{G(u_i,u_j)} $$
where $U_n$ denotes uniform distribution.
I think by putting weights $\lambda_{ij} \in [0,1]$ for each term $G(u_i,u_j)$ in the above equation, we can get a result of certain distribution. (Here we need to normalize the summation of $\lambda_{ij}$ to ensure that the summation equals to $C_N^2$). For example, we take $\lambda_{12}=0.5$ while other weights remain $1$. Thus, in the final result, $u_1$ and $u_2$ should be closer than other pairs. My question is, can we prove that (even in higher dimensions, not only on a unit circle)?
For the proof of uniform distribution mentioned at the beginning, you may refer to Proposition 4.4.1, Theorem 6.2.1, and Corollary 6.2.2 in Borodachov, S. V., Hardin, D. P., and Saff, E. B. Discrete energy on rectifiable sets. Springer, 2019.
