Grassmann Inequalities and Extremal Varieties in $${\mathbb {P}}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) $$

In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), et al. (J Optim Theory 169(1):1–16, which defines extremal varieties \(\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) \), we define a more general concept real Grassmannian \({G_p}\left( {{\mathbb \) ^p}{\mathbb \). This is based on minimization sums squares quadratic Plücker relations defining variety as well reverse maximisation problem. Such problems set inequalities Grassmann matrices, are essential for definition its dual variety. We prove these apply existing results, cases \({{ \wedge {R}^{2n}}}\) ^n}{\mathbb {R}^{2n}}}\). The resulting underline fact was demonstrated (2016, Linear Algebra 461:139–162, 2014), that such represented by multi-vectors acquire property unique singular value with total multiplicity. Crucial to numbers \(M_{n,p}\), calculated within mentioned above.