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

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