During my talk I will discuss the following result: for a continuous
function in Euclidean domain the following conditions are equivalent:
to be a viscosity solution to the $p$-Laplace equation, and
2) to possess
an asymptotic $p$-mean value property in the viscosity sense.
explain the proof for $p \in (1,\infty]$ and show why the proof does not
work in the case $p=1$. Moreover, I will discuss a generalization to the
setting of Carnot-Caratheodory groups. The Euclidean part of the talk is
based on Ishiwata-Magnanini-Wadade, Calc. Var. (2017), the second part
is based on joint work with Adamowicz, Pinamonti and Warhurst.