Spatial concavity properties of non-negative weak solutions of the filtration equations with absorption u t = (φ( u )) xx −ψ( u ) in Q = R ×(0, ∞), φ′[ges ]0, ψ[ges ]0 are studied. Under certain assumptions on the coefficients φ, ψ it is proved that concavity of the pressure function is a consequence of a ‘weak’ convexity of travelling-wave solutions of the form V ( x , t ) = θ( x −λ t + a ). It is established that the global structure of a so-called proper set [Bscr ] = { V } of such particular solutions determines a property of B -concavity for more general solutions which is preserved in time. For the filtration equation u t = (φ( u )) xx a semiconcavity estimate for the pressure, v xx [les ]( t +τ) −1 θ″(ξ), due to the B -concavity of the solution to the subset [Bscr ] of the explicit self-similar solutions θ( x /√( t +τ)) is proved. The analysis is based on the intersection comparison based on the Sturmian argument of the general solution u ( x , t ) with subsets [Bscr ] of particular solutions. Also studied are other aspects of the B -concavity/convexity with respect to different subsets of explicit solutions.