Smoothness of subspace sections of the unit balls of C(Q) and L1

We show that, for any integer n >= 2, the space C(Q) (where Q is a Hausdorff compact set, card Q > n) contains an n-dimensional subspace such that any translation thereof by a vector p, parallel to p parallel to < 1, intersects the unit ball B of C(Q) in a nonsmooth set. In L-1[0, 1], we show that if l is an arbitrary finite-dimensional subspace in L-1[0, 1], dim l. 1, then there exists a dense set in the unit ball B. L-1[0, 1] set of its translations that intersect the unit ball of L-1[0, 1] in smooth sets. As an application, we show that in L-1[0, 1] any finite-dimensional sun is convex. This extends the classical P.Orno-Yu. A. Brudnyi-E. A. Gorin's theorem to the effect that in L-1[0, 1] any Chebyshev set is either a singleton or is infinite-dimensional. (C) 2021 Elsevier Inc. All rights reserved.