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We characterise in this work the $q$-plurisubharmonic functions in terms ofthe theory of viscosity solutions. We show that an upper semicontinuousfunction is $q$-plurisubharmonic if and only if its complex Hessian has at most$q$ strictly negative eigenvalues in the viscosity sense. This characterisationis then used to prove that the supremum convolution of a (strictly)$q$-plurisubharmonic function is again (strictly) $q$-plurisubharmonic on amaybe different set of definition. Finally, we use the supremum convolution todeduce a new characterisation for the $q$-pseudoconvex subsets in$\mathbb{C}^n$.

On convex hulls and pseudoconvex domains generated by $q$-plurisubharmonic functions, part III