On a Banach space we call a
projection, P, bicontractive, if ∥P∥≦ 1 and ∥I − P∥≦ 1. In this paper we
completely describe bicontractive projections on an Lp-space (1 ≦ p < ∞) by
showing that for every such bicontractive projection P, 2P − I is an involutive
linear isometry. Duality then gives the same result for pre-dual L1-spaces (in
particular for M-spaces). The analysis of bicontractive projections is used,
with p≠2, to describe all Banach lattices which are linearly isometric to an
Lp-space.