[For part II, cf. the review below, Zbl 0646.53009).] This is an interesting paper about equiaffine Weingarten surfaces and hypersurfaces continuing investigations of \textit{A. Švec} [Czech. Math. J. 37, 567-572 (1987)], \textit{R. Schneider} [Math. Z. 101, 375-406 (1967; Zbl 0156.201)] and \textit{A. Schwenk} and the reviewer [Arch. Math. 46, 85-90 (1986; Zbl 0563.53009)]. The author proves a series of uniqueness results about compact locally strongly convex hypersurfaces (with or without boundary). Main tool for the proof are the equiaffine Codazzi equations as partial differential equations (index method, integral formulas). [The reviewer would like to point out that the author meanwhile was able to generalize some of the results of this paper further (to appear).]