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The answer is "no". In the plane, the circle of diameter greater than $1$ but smaller than $\sqrt2$ and centered at $({1\over2}, {1\over2})$ contains no point with integer coordinates, and the square circumscribed about it with a horizontal diagonal contains no such point either. Similar examples exist in every dimension, with a suitable $n$-sphere instead of a circle and the cross-polytope with diameters parallel to the coordinate axes, circumscribed about the sphere in place of the square. The diameter of the $n$-sphere can be very close to $\sqrt n$, and the number of vertices is $2n$.