Motivated by earlier considerations of interval interpolation problems as w
ell as a particular application to the reconstruction of railway bridges, w
e deal with the problem of univariate convexity preserving interval interpo
lation. To allow convex interpolation, the given data intervals have to be
in (strictly) convex position. This property is checked by applying an abst
ract three-term staircase algorithm, which is presented in this paper. Addi
tionally, the algorithm provides strictly convex ordinates belonging to the
data intervals. Therefore, the known methods in convex Lagrange interpolat
ion can be used to obtain interval interpolants. In particular, we refer to
methods based on polynomial splines defined on grids with additional knots
.