Vertexwise criteria for admissibility of alcoves

We give a new description of the set $Adm(\mu)$ 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.