  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.
