In the paper we examine properties of strongly unique best approximation in terms of extremal functionals in an abstract normed linear space. Using some new or modified tools (e.g., the shadow of a set, strongly tangent sets, I-sets) we express criteria of strong uniqueness both in linear and nonlinear approximation. Some further remarks are contributed to the discussion on Poreda's problem concerning the behavior of the strong unicity constants and a detailed description of properties of the tangent cone of Dubovitsky-Milyutin is given.