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:
https://mathoverflow.net/questions/298898/an-inequality-related-to-lagranges-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.

2. 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)

3. As users @fedja and @mahdi suggested in https://mathoverflow.net/questions/298898/an-inequality-related-to-lagranges-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!