  We prove an abstract result ensuring that one-sided geometric control yields
two-sided estimates for functions satisfying general conditions. Our findings
resonate in the context of nonlinear elliptic problems, including
supersolutions to fully nonlinear elliptic equations and functions in the De
Giorgi class. Among the consequences of our abstract results are regularity
estimates, and conditions for a continuous function to be in the class of
viscosity solutions. We also prove that one-sided geometric control yields
$L^pL^\infty$-estimates. It provides a converse to the implication in the De
Giorgi-Nash-Moser theory.
