convexity of product of squared Euclidean distances with common point $i$, $||x_i-x_j||||x_i-x_k||$?

Question

convexity of product of squared Euclidean distances from common point $i$ to points $j,k$ given by $||x_i-x_j||^2||x_i-x_k||^2$ holds true? Powers of Euclidean distance are convex. Would it change for specific powers $p,q$ such as $||x_i-x_j||^p||x_i-x_k||^q$?

$p,q$ could be fractions as well. $x_i,x_j,x_k \in \mathbb{R^d}$.

Answer 1

Such a function cannot be convex. This function is non-negative, it is zero on the union of two subspaces $\{x: \ x_i = x_j\}$ and $\{x: \ x_i = x_k\}$. The union of two subspaces is non-convex in general.