I have the following function $$H(\mathbf{x}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}|+\cdots+a_{n}|x_{n}-A_{n}| +b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}|+\cdots+a_{n}|x_{n}-B_{n}| +c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+\cdots+c_{n}|x_{n}-x_{n-1}|+c_{n+1}|x_{n+1}-x_{n}|$$ where all coefficients $a_{1}, a_{2}, \cdots, b_{1}, b_{2}, \cdots, A_{1}, A_{2}, \cdots, B_{1}, B_{2}, \cdots, c_{1}, \cdots$, $x_{0}$ and $x_{n+1}$ are positive constants.
For two dimensional case, I can plot $$H(x_{1},x_{2}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}| +b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}| +c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+c_{3}|x_{3}-x_{2}|$$ and it indeed has only one minimum,
However, I cannot prove it in a more general case. Does anyone once had encountered such problems?