A Steiner formula in the L Brunn Minkowski theory

Abstract

We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases.Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed Lp affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given. *

$L_p$-Steiner quermassintegrals

“…It was also shown in [29] that for p = 0 the above formula gives the classical Steiner formula (1). In particular, the sum appearing in ( 2) is then a finite sum.…”

“…In [29] an L p Steiner formula was proved for the L p affine surface area. Namely, if K is C 2 + , then we have for all suitable t and for all p ∈ R, p = −n, that…”

“…In particular, the sum appearing in ( 2) is then a finite sum. The sum is also finite for all p = − n(l−1) l , l ∈ N, [29]. In general, the sum is infinite and thus proving (2) requires a careful analysis of convergence issues.…”

“…It was shown in [29] that the following L p Steiner formulas hold. They are a generalization of the classical Steiner formula, which corresponds to the case p = 0 and Lutwak's Steiner formula of the dual Brunn Minkowski theory, which corresponds to the case p = ±∞:…”

“…In the L p Steiner formulas (15), the first coefficient is V p 0 (K) = as p (K). It was shown in [29] that when p = −n(l−1) l with l ∈ N, the sums (15) are finite with the highest term t n(2l−1) :…”