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 \(\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) \), we define a more general concept of extremal varieties of the real Grassmannian \({G_p}\left( {{\mathbb {R}^n}} \right) \) in \(\mathbb {P}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) \). 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 \({{ \wedge ^2}{\mathbb {R}^{2n}}}\) and \({{ \wedge ^n}{\mathbb {R}^{2n}}}\). 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 \(M_{n,p}\), which are calculated within the cases mentioned above.

为了继续在Leventides和Petroulakis工作（Adv Appl Clifford Algebras 27：1503-1515，2016），Leventides等。（J Optim Theory Appl 169（1）：1-16，2016），它定义了\（\ mathbb {P} \ left（{{\ bigwedge ^ 2} {\ mathbb {R} ^ n}} \右）\） ，我们定义的极值品种真实格拉斯曼的更一般概念\（{G_P} \左（{{\ mathbb {R} ^ N}} \右）\）在\（\ mathbb {P} \ left（{{\ bigwedge ^ p} {\ mathbb {R} ^ n}} \ right）\）。这个概念基于最小化定义格拉斯曼变种的二次普吕克关系的平方和以及逆最大化问题。这样的极值问题在一组Grassmann矩阵上定义了一组Grassmannian不等式，这对于定义Grassmann变体及其对偶极值变体是必不可少的。我们定义了并证明了一般格拉斯曼式的这些不等式，并在\（{{\ wedge ^ 2} {\ mathbb {R} ^ {2n}}} \）和\（{{ ^ n} {\ mathbb {R} ^ {2n}}} \）。最终的极端变种突出了这一事实，这在Leventides等人的研究中得到了证实。（2016，Linear Algebra Appl 461：139–162，2014），这种变体是由多个向量表示的，这些向量获得具有唯一多重性的唯一奇异值的属性。对于这些不等式至关重要的是数字\（M_ {n，p} \），它是在上述情况下计算得出的。