Before revision:   We show that  there is no polynomial-time (\frac{3{2}-\varepsilon)-approximation algorithm for any \varepsilon~>~0 }  unless \mathsf{P}=\mathsf{NP}.
After revision:  In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. We show that there is no polynomial-time (\frac{3{2}-\varepsilon)-approximation algorithm for any \varepsilon~>~0 } unless \mathsf{P}=\mathsf{NP}.