This paper deals with lower bounds on the approximability of different subproblems of the Traveling Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general (unless $\mathcal{P}=\mathcal{NP}$). First of all, we present an improved lower bound for the Traveling Salesman Problem with Triangle Inequality, Delta-TSP for short. Moreover our technique, an extension of the method of Engebretsen [11], also applies to the case of relaxed and sharpened triangle inequality, respectively, denoted $\Delta_\beta$-TSP for an appropriate β. In case of the Delta-TSP, we obtain a lower bound of $\frac{3813}{3812}-\varepsilon$ on the polynomial-time approximability (for any small $\varepsilon> 0$), compared to the previous bound of $\frac{5381}{5380}-\varepsilon$ in [11]. In case of the $\Delta_\beta$-TSP, for the relaxed case ($\beta> 1$) we present a lower bound of $\frac{3803+10\beta}{3804+8\beta}-\varepsilon$, and for the sharpened triangle inequality ($\frac{1}{2}< \beta< 1$), the lower bound is $\frac{7611+10\beta^2+5\beta}{7612+8\beta^2+4\beta}-\varepsilon$. The latter result is of interest especially since it shows that the TSP is $\mathcal{APX}$-hard even if one comes arbitrarily close to the trivial case that all edges have the same cost.