  We establish sharp $W^{2,p}$ regularity estimates for viscosity solutions of
fully nonlinear elliptic equations under minimal, asymptotic assumptions on the
governing operator $F$. By means of geometric tangential methods, we show that
if the {\it recession} of the operator $F$ -- formally given by
$F^*(M):=\infty^{-1} F(\infty M)$ -- is convex, then any viscosity solution to
the original equation $F(D^2u) = f(x)$ is locally of class $W^{2,p}$, provided
$f\in L^p$, $p>d$, with appropriate universal estimates. Our result extends to
operators with variable coefficients and in this setting they are new even
under convexity of the frozen coefficient operator, $M\mapsto F(x_0, M)$, as
oscillation is measured only at the recession level. The methods further yield
BMO regularity of the hessian, provided the source lies in that space. As a
final application, we establish the density of $W^{2,p}$ solutions within the
class of all continuous viscosity solutions, for generic fully nonlinear
operators $F$. This result gives an alternative tool for treating common issues
often faced in the theory of viscosity solutions.
