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We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions. 2010 Mathematics Subject Classification. 47J10, 39B62, 35B50. Key words and phrases. Nonlinear eigenvalue problems; uniqueness ofdoi:10.2996/kmj/1414674621 fatcat:b45s5w3dpzbwlbyrr443zuyk34