Suppose $\{X\}_{i=1}^n\overset{i.i.d}{\sim}X$, and $X\in\mathbb{R}^d$ has density,$$f_{\theta}(x)=c\exp\left\{-||x-\theta||\right\},\theta\in\mathbb{R}^d,$$ where $||\cdot||$ denotes the Euclidean norm.
Show that the MLE $\hat{\theta}$ exists but is not unique when $n$ is even.
I know how to prove that the MLE $\hat{\theta}$ exists, by noticing $\underset{\theta\rightarrow\partial\mathbb{R}^d}{\lim} \log f(\theta)=-\infty$, however I don't know how to prove it is not unique. I understand that the log-likelihood function, $l(\theta)=-\sum_{i=1}^n||x_i-\theta||$ is a convex, but not strictly convex, function, but this does not guarantee the nonuniqueness. I also tried taking the first and second gradient of $l(\theta)$. But even I have a non-negative definite $l(\theta)$, I still don't have the non-uniqueness...
---Update---
The original question is here. This is from Mathematical Statistics (Bickel).
The minimizer of such set of points is given by the Geometric Median.
For example, for $ d = 1 $ you get the Median which is not unique for an odd set of different numbers.
For higher dimensions you need to take care of the case the points are collinear, which basically means that the problem, is again, equivalent to 1D.