2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143293We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.v1: original version (>20 pages) v2: v1 was far too verbose; this pares it to under 10 pagesProbabilityCombinatorics60J10text