  Inspired by an $L_p$ Steiner formula for the $L_p$ affine surface area proved
by Tatarko and Werner, we define, in analogy to the classical Steiner formula,
$L_p$-Steiner quermassintegrals. Special cases include the classical mixed
volumes, the dual mixed volumes, the $L_p$ affine surface areas and the mixed
$L_p$ affine surface areas. We investigate the properties of the $L_p$-Steiner
quermassintegrals in a special class of convex bodies. In particular, we show
that they are rotation and reflection invariant valuations in this class of
convex bodies with a certain degree of homogeneity. Such valuations seem new
and have not been observed before.
