Let \(\Gamma\) be a hexagonic Lie incidence structure, e.g.\ a long root geometry of a (thick irreducible) spherical Tits building; compare \textit{E. E. Shult} [Points and lines. Characterizing the classical geometries. Berlin: Springer (2011; Zbl 1213.51001)]. The authors classify all convex subspaces of \(\Gamma\) under the assumption that \(\Gamma\) contains polar spaces of rank \(3\): then every convex subspace of \(\Gamma\) either corresponds to a residue of a flag in the associated spherical building, or it originates from ``special'' pairs of points (i.e.\ non-collinear pairs of points that are collinear to a unique common point). Moreover, similar classification results are proved for many other Lie incidence geometries, in particular for all projective and polar Grassmannians, and for exceptional Grassmannians of diameter at most \(3\); this generalizes Corollary 6.3 of [\textit{A. Kasikova}, Adv. Geom. 9, No. 4, 541--561 (2009; Zbl 1183.51004)] and Corollary 4.2 of [\textit{M. Pankov}, Bull. Belg. Math. Soc. - Simon Stevin 19, No. 2, 345--366 (2012; Zbl 1247.51007)].