We show that, for any integer n≥2, the space C(Q) (where Q is a Hausdorff compact set, cardQ>n) contains an n-dimensional subspace such that any translation thereof by a vector p, ‖p‖<1, intersects the unit ball B of C(Q) in a nonsmooth set. In L1[0,1], we show that if ℓ is an arbitrary finite-dimensional subspace in L1[0,1], dimℓ⩾1, then there exists a dense set in the unit ball B⊂L1[0,1] set of its translations that intersect the unit ball of L1[0,1] in smooth sets. As an application, we show that in L1[0,1] any finite-dimensional sun is convex. This extends the classical P. Ørno–Yu.A. Brudnyi–E.A. Gorin’s theorem to the effect that in L1[0,1] any Chebyshev set is either a singleton or is infinite-dimensional.