The attainment set of the $\varphi-$envelope of a function at a given point is investigated. The inclusion of the attainment set of the $\varphi$-envelope of the closed convex hull of a function into the attainment set of the function is preserved in sufficiently general settings toencompass the case $\varphi $ being a norm in a power not less than $1$. The nonemptiness of the attainment set is guaranteed on genericsubsets of a given space, in several fundamental cases.