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 are positive.

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?