We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb
R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball
B_r^d \subset {\mathbb 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
{\mathbb R}^d. Since such a line has length 1, the diameter of the embedding
ball must exceed 1.