This paper investigates the connections among LYM-type and Bollobás type inequalities concerning the Sperner, the intersecting Sperner, the non-union Sperner, the complement-free Sperner and finally the non-union, intersecting Sperner polytops. (For a systematic introduction to the theory of the convex hull of set systems and the complete description of those polytops above, see \textit{A. Derbala} and \textit{K. Engel} [Algorithmic investigations of heighted extremal set problems, Proc. Conf. Extremal Set Theory, Visegrád, Hungary, 1991]. The main tool is a generalization of Daykin's LYM-extension [\textit{D. E. Daykin}, Antichains of subsets of a finite set, Graph theory and combinatorics, Proc. Conf. Hon. P. Erdős, Cambridge 1983, 99-106 (1984; Zbl 0547.05002)]. The unreferred Proposition 6 (which is stated to be the most attractive result of the paper) is coincident with the Remark after Lemma 1 in \textit{K. Engel} and \textit{P. L. Erdős} [Sperner families satisfying additional conditions and their convex fulls, Graphs Comb. 5, No. 1, 47- 56 (1989; Zbl 0709.05038)].