Asymptotic theory of $C$-pseudo-cones

In this paper, we study the non-degenerated $C$-pseudo-cones which can be uniquely decomposed into the sum of a $C$-asymptotic set and a $C$-starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional $T_{\mathrm{\Theta }}(E)$ of the non-degenerated $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_{\mathrm{\Theta }}(E)$, we establish a decay estimate for $T_{\mathrm{\Theta }}(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_{\mathrm{\Theta }}(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones.