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:

- 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: $3^{1 - \frac{1}{p}}$

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)

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!