On the largest multilinear singular values of higher-order tensors

Let $\sigma_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$\sigma_1^2+\dots+\sigma_{n-1}^2+\sigma_{n+1}^2+\dots+\sigma_{N}^2\leq (N-2)\|\mathcal T\|^2 + \sigma_n^2,\quad n=1,\dots,N, \qquad\qquad (1)$$ where $\|\cdot\|$ denotes the Frobenius norm. We also show that at least for the cubic tensors the inverse problem always has a solution. Namely, for each $\sigma_1,\dots,\sigma_N$ that satisfy (1) and the trivial inequalities $\sigma_1\geq \frac{1}{\sqrt{I}}\|\mathcal T\|,\dots, \sigma_N\geq \frac{1}{\sqrt{I}}\|\mathcal T\|$, there always exists an $I\times \dots\times I$ tensor whose largest multilinear singular values are equal to $\sigma_1,\dots,\sigma_N$. For $N=3$ we show that if the equality $\sigma_1^2+\sigma_2^2= \|\mathcal T\|^2 + \sigma_3^2$ in (1) holds, then $\mathcal T$ is necessarily equal to a sum of multilinear rank-$(L_1,1,L_1)$ and multilinear rank-$(1,L_2,L_2)$ tensors and we give a complete description of all its multilinear singular values. We establish a connection with honeycombs and eigenvalues of the sum of two Hermitian matrices. This seems to give at least a partial explanation of why results on the joint distribution of multilinear singular values are scarce.