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It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter's formula predicts linear dependence of the buckling load $\lambda(h)$ of the shell thickness $h$ ($h>0$ is a small parameter), one observes the dependence $\lambda(h)\sim h^{3/2}$ in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in [\textit{J. Nonl. Sci.,} Vol. 26, Iss. 1, pp. 83--119, Feb. 2016], that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling $\lambda(h)\sim h^{5/4},$ while shape imperfections are likely to result in the scaling $\lambda(h)\sim h^{3/2}.$ In this work we prove, that in fact the buckling load $\lambda(h)$ of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then $\lambda(h)\sim h,$ and when the the cross section curve has positive curvature except at finitely many points, then $C_1h^{8/5}\leq \lambda(h)\leq C_2h^{3/2}$ for $h$ small thickness $h>0.$, Comment: 28 pages, 4 figures