What is the relationship between the definitions of $s$-capacity (page 13 [here][1]) and $p$-capacity ([here][2]) of a set? 

Are they equivalent? If not, what inequalities hold? What is the difference (in terms of applications) between them?


  [1]: https://www.math.unipd.it/~monti/PAPERS/Frequency2012.pdf
  [2]: https://math.stackexchange.com/questions/376928/the-definition-of-p-capacity-of-a-set-a-subset-mathbbrn