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Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that

$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\ a_j & b_j & c_j \\ a_k & b_k & c_k \end{matrix}\right)\right|^p \right)^{\frac{1}{p}} \leq c_p \left( \sum_{i=1}^n |a_i|^p \right)^{\frac{1}{p}} \left( \sum_{1\leq j <k \leq n} \left| \det\left(\begin{matrix} b_j & c_j \\ b_k & c_k \end{matrix}\right)\right|^p \right)^{\frac{1}{p}} $$

where $c_p = \max(1, 3^{1-\frac{2}{p}})$.

A 2-dimensional analogue of this problem was discussed here: An inequality related to Lagrange's identity and $L_p$ norm

Remark:

  1. When $p=1$, the proof is straightforward since $$\left| \det\left(\begin{matrix} a_i & b_i & c_i \\ a_j & b_j & c_j \\ a_k & b_k & c_k \end{matrix}\right)\right| \leq |a_i| \left| \det\left(\begin{matrix} b_j & c_j \\ b_k & c_k \end{matrix}\right)\right| + |a_j| \left| \det\left(\begin{matrix} b_i & c_i \\ b_k & c_k \end{matrix}\right)\right| +|a_k| \left| \det\left(\begin{matrix} b_i & c_i\\ b_j & c_j \end{matrix}\right)\right| $$, by Laplace expansion and triangle inequality. Summing up all these inequalities is enough. $p = \infty$ case can be proved in a similar way.

Using Holder's inequality directly on the Laplace expansion gives a weaker bound: $3^{1 - \frac{1}{p}}$

  1. When $p=2$, LHS is the volume of Parallelepiped spanned by three vectors $a,b,c$, while RHS is the norm of $a$ times the area of parallelogram spanned by $b,c$, so the inequality is clearly true. (This fact can be proved by using Cauchy-Binet)

  2. As users @fedja and @mahdi suggested in An inequality related to Lagrange's identity and $L_p$ norm , this problem is closely related to Riesz-Thorin interpolation theorem. However, I find it difficult to apply the theorem directly on my problem.

Thanks!

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  • $\begingroup$ If we could prove that $$ \sum_{1\leq i,j,k\leq n} |a_i c_{jk}-a_jc_{ik}+a_kc_{ij}|^2 \leq \left(\sum_{i=1}^n a_i^2 \right) \left( \sum_{1\leq i,j\leq n}c_{ij}^2 \right) $$ for all $a \in \mathbb R^n$, and all $n\times n$ skew-symmetric matrix $C=[c_{ij}]$, then Riesz-Thorin interpolation theorem implies your mentioned inequality. $\endgroup$ – Mahdi Jul 8 '18 at 10:03
  • $\begingroup$ Yes I have already proved that too. I am a bit busy these days, and will update a proof later. Thank you Mahdi! $\endgroup$ – Chen Dan Jul 9 '18 at 1:56
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Actually, Riesz-Thorin interpolation theorem proves a generalized result. That is,

Observation. for all $a \in \mathbb C^n$, and all $n\times n$ skew-Hermitian matrix $C$, we have $$ \sum_{1\leq i<j<k\leq n} |a_i c_{jk}-a_jc_{ik}+a_kc_{ij}|^p \leq c^p_p\left(\sum_{i=1}^n a_i^p \right) \left( \sum_{1\leq i<j\leq n}c_{ij}^p \right) $$ where $c_p := \max (1,3^{1-2/p})$ and $1\leq p \leq \infty$.

When $p=1,\infty$, above observation is obvious. Also, above observation holds true when $p=2$ (as OP said). Now, by considering a multilinear function $\Lambda$ defined as $[\Lambda(a,C)]_{i,j,k} = a_i c_{jk}-a_jc_{ik}+a_kc_{ij}$ and interpolating between $1$ and $2$ and then between $2$ and $\infty$, above observation was proved for $1\leq p \leq \infty$ (similar to the answer of a previous question of OP).

If we set $c_{ij} := \det \begin{pmatrix}b_i&c_i\\b_j&c_j\end{pmatrix}$ in the above observation, the mentioned inequality of OP follows.

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