# Kernel with respect mean distances on a unit sphere

I am trying to understand a proof by G.Wagner in his paper "On Means on Distances on the Surface of a Sphere (Lower Bounds)": http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102645739

I want to essentially simplify the proof of Theorem 2 for the case of the unit sphere in 3 dimensional space when $\alpha =-1$.

The Theorem states:

$$E_{\alpha} \geq -c(\alpha,d) \cdot N^{1-\alpha/(d-1)} \ \ \ \ \ (1-d < \alpha < 3-d)$$

Where $d=3$ is the dimension, $N$ is the number of points distributed on the sphere and $c$ is a constant.

In the proof, it then says: "In the case of an unbounded kernel $k_{\alpha}$, we proceed in a different way. Together with the kernel $k_{\alpha}(\theta) = (2-2cos(\theta_{1}))^{\alpha /2}$ consider the more general kernel $$d_{r} (cos(\theta _{1}))=(r+\frac{1}{r}-2cos(\theta _{1}))^{\alpha /2} \ \ \ \ \ (0 < r \leq 1)$$

My issue is I don't really understand what the "kernel" means in this context and so can't really progress with the understanding the proof, I can therefore only regurgitate it. Could someone please explain what this means in the context of the theorem/paper, I appreciate it is hard to understand what I am trying to explain here, therefore I have included a link to the paper for clarification.

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