Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS_rⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys.16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS_rⁿ. The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys.8:478 (1967)] and the underlying symmetry property.