Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D_m in the Euclidean space R^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B^d_r⊂.ℝ^d of radius r<1/2 unless one of two conditions is met: (1) the embedding manifold has dimension d≥2m; (2) the embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold ℝ^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.