A kind of higher-dimensional geometric structure and Malan conjecture. III (Q2724931)

[For part I and II see \textit{Q.~Luo}, ibid. 22, No.~4, 1-8 (1996; Zbl 0872.15013) and \textit{B.~Tao} and \textit{Q.~Luo}, ibid. 25, No.~3, 1-5 (1999; Zbl 0959.15014)]NEWLINENEWLINENEWLINELet \(S_N\) be the unit sphere in an \(n\)-dimensional real vector space, let \(I_j=(1,\lambda_1,\dots,\lambda_{n-1})\) be all possible vectors with \(|\lambda_i|=1\). The Malan conjecture is equivalent to prove that \(\min_{A\in S_n}(\sum_{j=1}^{2^{n-1}}|\langle I_j,A\rangle|)/2^{n-1}=1/\sqrt 2\). In the previous parts, the authors claimed that the function \(f(A)\) reaches its minimal value in an intersection point of domains. In this paper, the authors give the geometric interpretation of these conceptions for dimension \(n\leq 5\).