A finite metric space $(X,d)$ is of negative type if $d$ satisfies $$ \sum_{x\in X} c_x=0 \implies \sum_{x,y\in X} c_x c_y d(x,y)\leq 0 $$ - def from [M. DEZA AND H. MAEHARA, METRIC TRANSFORMS AND EUCLIDEAN EMBEDDINGS]
Now, given a finite metric space $(X=\{x_1,\dots, x_m\}, d)$, for any values $c_1, \dots, c_m$ such that $$\sum_{i=1}^m c_i=0,$$ I can consider the constant $C= \sum_{k=1}^m c_k d(x_k, x_1)$, and using the triangle inequality and the simmetry of the metric $d$ I have $$ \sum_{i,j=1}^m c_i c_j d(x_i,x_j)\leq\sum_{i,j=1}^m c_i c_j (d(x_i,x_1)+d(x_1,x_j))=\sum_{i,j=1}^m c_i c_j d(x_i,x_1)+\sum_{i,j}^m c_i c_jd(x_j,x_i) $$ $$ =C\sum_{i=1}^m c_i+ C\sum_{j=1}^m c_j =0 $$
thus $(X,d)$ satisfies the condition of negative type.
Something is wrong in the previous proof but I cannot understand where is the error! It seems to me that any finite metric space is of a negative type, but extensive literature had been studied transformations of finite metric space to make them of negative type... Anyone can help me in clarifying this matter?
You are mulitiplying triangle inequality by numbers which are not all positive and adding then up; note that some of the $c_i$ 's are necessarily negative.