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Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially ($S$_{$r$}) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition ($S$_{$r$}). Let Δ be a ($d$−1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let ($h$_{$i$,$j$}(Δ))_{0≤$j$≤$i$≤$d$}be the$h$-triangle of Δ and ($h$_{$i$,$j$}(Γ(Δ)))_{0≤$j$≤$i$≤$d$}be the$h$-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially ($S$_{$r$}) and for every$i$and$j$with 0≤$j$≤$i$≤$r$−1, the equality$h$_{$i$,$j$}(Δ)=$h$_{$i$,$j$}(Γ(Δ)) holds true.

A classification of $\SLn$ contravariant, continuous function-valued valuations on convex bodies is established. Such valuations are natural extensions of $\SLn$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.

The centroaffine Minkowski problem is studied, which is the critical case of the <i>L</i>_<i>p</i>-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions. In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.