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In this paper, for the development of the study of almost Gorenstein gradedrings, we discuss some relations between almost Gorensteinness ofCohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for aCohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ tobe almost Gorenstein in terms of the $h$-vector of $R$ and we also characterizealmost Gorenstein homogeneous domains with small socle degrees in terms of the$h$-vector of $R$. Moreover, we also provide the examples of almost Gorensteinhomogeneous domains arising from lattice polytopes.

Almost Gorenstein homogeneous rings and their $h$-vectors