  In this paper we discuss convexity, its average principle, an extrinsic
average variational method in the Calculus of Variations, an average method in
Partial Differential Equations, a link of convexity to $p$-subharmonicity,
subsolutions to the $p$-Laplace equation, uniqueness, existence, isometric
immersions in multiple settings. In particular, we show that a convex function
on $\mathbb{R}^n$ is a $p$-subharmonic function, for every $p > 1$, and a $C^2$
convex function on a Riemannian manifold is a $p$-subharmonic function $f$, for
every $p > 1\, .$ We also show that a $C^2$ convex function which is a
submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p
\ge 1\, .$ This result is sharp. As further applications, via function growth
estimates in $p$-harmonic geometry, we prove that every $p$-balanced
nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold
is constant for $p > 1$. In particular, every $L^q$, nonnegative, convex
function of class $C^2$ on a complete noncompact Riemannian manifold is
constant for $q > p -1 > 0\, .$
