Is the product of distances to a point convex?

Question

Let the distance from point $a$ to point $x_i$ be the $L_2$ norm,

\begin{align} \|a - x_i\|_2 \end{align}

This is a convex function in $a$.

Are the following functions convex in $a$ on $a \geq 0$?

\begin{align} \Pi_{i=1}^N \|a - x_i\|_2 \end{align}

and

\begin{align} a * \Pi_{i=1}^N \|a - x_i\|_2 \end{align}

And why?

Answer 1

Take $N=2$, $x_1=1,x_2 = -1$, then for $|a| \le 1$ we have $|a-1||a+1| = |a^2-1| = 1-a^2$ which is not a convex function of $a$.