Power-law asymptotics of fractional $L^p$ polar projection bodies

The notion of $s$--fractional $L^p$ polar projection bodies, recently introduced by Haddad and Ludwig (Math.\ Ann.\ \textbf{388}:1091--1115, 2024), provides a bridge between fractional Sobolev theory and convex geometry. In this manuscript, we study the limit of their Minkowski gauges under two natural asymptotic regimes: \\ \hspace*{3em} (a) first sending $p \to \infty$ and then $s \to 1^-$; \\ \hspace*{3em} (b) first sending $s \to 1^-$ and then $p \to \infty$. \\ Our main result shows that these two limiting processes commute. As a consequence, we derive precise asymptotic behavior for the associated volumes and dual mixed volumes, thereby linking the (fractional) $L^p$ polar projection bodies to the newly introduced (fractional) $L^\infty$ polar projection bodies. These results further yield new geometric inequalities, including endpoint Lipschitz/Hölder isoperimetric--type and variants of Pólya--Szegő inequalities in the $L^\infty$ setting.