Vertexwise criteria for admissibility of alcoves

We give a new description of the set ${\rm Adm}(\mu)$ of admissible alcoves as an intersection of certain ``obtuse cones'' of alcoves, and we show this description may be given by imposing conditions vertexwise. We use this to prove the vertexwise admissibility conjecture of Pappas-Rapoport-Smithling. The same idea gives simple proofs of two ingredients used in the proof of the Kottwitz-Rapoport conjecture on existence of crystals with additional structure.