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We prove an analogue of the classical Steiner formula for the $L_p$ affinesurface area of a Minkowski outer parallel body for any real parameters $p$. Weshow that the classical Steiner formula and the Steiner formula of Lutwak'sdual Brunn Minkowski theory are special cases of this new Steiner formula. Thisnew Steiner formula and its localized versions lead to new curvature measuresthat have not appeared before in the literature. They have the intrinsicvolumes of the classical Brunn Minkowski theory and the dual quermassintegralsof the dual Brunn Minkowski theory as well as special cases. Properties ofthese new quantities are investigated, a connection to information theory amongthem. A Steiner formula for the $s$-th mixed $L_p$ affine surface area of aMinkowski outer parallel body for any real parameters $p$ and~$s$ is alsogiven.

A Steiner formula in the $L_p$ Brunn Minkowski theory