$L_p$-Steiner quermassintegrals

Abstract

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

Curvature functionals on convex bodies

“…When in addition k = m, we get Proof The proof follows immediately from results in [35]. We present an outline of the proof for completeness.…”

“…where the H j are the j-th normalized elementary symmetric functions of the principal curvatures. The c(n, p, m) are certain binomial coefficients, see [34,35] for the details. The L p Steiner coefficients were studied in [35], where it was proved, among other results, that they are valuations on the set of convex bodies.…”

Curvature functionals on convex bodies

“…3 Properties of the weighted 𝐿 𝑝 -affine surface areas Proof The proof follows immediately from results in [35]. We present an outline of the proof for completeness.…”

“…where the 𝐻 𝑗 are the 𝑗-th normalized elementary symmetric functions of the principal curvatures. The 𝑐(𝑛, 𝑝, 𝑚) are certain binomial coefficients, see [34,35] for the details.…”

“…The 𝐿 𝑝 Steiner coefficients were studied in [35], where it was proved, among other results, that they are valuations on the set of convex bodies.…”