Convex Functions are $p$-Subharmonic Functions, $p>1$ On ${\mathbb{R}}^n$ with Applications

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 $p>1\mathrm{.}$ We also show that a $C^2$ convex function which is a submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p\ge 1\mathrm{.}$ 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 $q>p-1>0\mathrm{.}$