I have a function $f(x)$ defined as follows $$f(x)=f_1(x_0,x_1)+f_2(x_0,x_2)+\cdots f_T(x_0,x_T)$$ each of the individual function $f_i(x_0,x_i)$ is non-negative and jointly concave over $(x_0,x_i)$ therefore the summation is also concave. If I fix $x_0=c$ then the new function becomes $$f_c(x)=f_1(c,x_1)+f_2(c,x_2)+\cdots f_T(c,x_T)$$ My question is as follows. Is $f_c(x)$ a concave function of $x_1,x_2 \cdots x_T$? Any help in this regard will be much appreciated. Thanks in advance.