  We prove that universal differentiability sets in Euclidean spaces possess
distinctive structural properties. Namely, we show that any universal
differentiability set contains a `kernel' in which the points of
differentiability of each Lipschitz function are dense. We further prove that
no universal differentiability set may be decomposed as a countable union of
relatively closed, non-universal differentiability sets. The sharpness of this
result, with respect to existing decomposibility results of the opposite
nature, is discussed.
