  Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in
computational physics and statistical computing. Yet, their most fundamental
properties, such as relations between convergence characteristics of their
various versions, are not well understood.
  In this paper we prove the solidarity of their spectral gaps: if any of the
random scan or $d!$ deterministic scans has a~spectral gap then all of them
have. Our methods rely on geometric interpretation of the Gibbs samplers as
alternating projection algorithms and analysis of the rate of convergence in
the von Neumann--Halperin method of cyclic alternating projections.
  In addition, we provide a quantitative result: if the spectral gap of the
random scan Gibbs sampler scales polynomially with dimension, so does the
spectral gap of any of the deterministic scans.
