Valuations on Polytopes Containing the Origin in Their Interiors

Abstract

We give a classification of non-negative or Borel measurable, SL(d) invariant, homogeneous valuations on the space of d-dimensional convex polytopes containing the origin in their interiors. The only examples are volume, volume of the polar body, and the Euler characteristic.

$L_p$ John Ellipsoids

“…Since its introduction in the mid 90 s, L p curvature (and the functionals it gives rise to) have attracted increased interest (see, for example, Campi and Gronchi [7,8], Chou and Wang [9], Gardner [11], Guan and Lin [15], Hu, Ma and Shen [16], Hug and Schneider [18], Klain [19], Ludwig [24,25,26], Meyer and Werner [38], Ryabogin and Zvavitch [43], Sch€ u utt and Werner [45,46], Stancu [47,48], Umanskiy [50], Werner [51], and also [17], [27,28,29], [32,34,35] and [30]). In this paper we will be interested in minimizing total L p -curvature of a body under SLðnÞ-transformations of the body: given a smooth convex body K in R n , that contains the origin in its interior, and a xed real p > 0, nd min 2SLðnÞ ð S nÀ1 f p ðK; uÞ dSðuÞ:…”

Centro-affine tensor valuations

“…The set of double pyramids will be denoted by R n and the set of straight double pyramids by Q n . In [24], Ludwig proved that if a real valued valuation µ : P n o → R vanishes on all SL(n)-images of elements in R n , then it vanishes on P n o . A componentwise application of this fact yields the following result.…”

“…However, one crucial part of the problem remained open since one of their assumptions was a certain behavior of the maps on convex polytopes. The first step to bridge this last gap had already been taken by Ludwig [24], but the complete result was only established very recently by the authors [16]: for all K ∈ K n o . Here, χ denotes the Euler Characteristic, V stands for volume, K * is the polar body of K, and the Ω ϕ are Orlicz affine surface areas.…”

Moments and valuations

“…The proof of the first assertion can be found in [20]. In order to establish the part on the homogeneity we follow [28]. Let µ be q-homogeneous, q ∈ R. By the first part of the theorem we obtain, for all s, t > 0,…”

“…, I n−1 , −c e ′ , de n is contained in the SL(n)-image of Q n . From relations(27),(28) and the definition of f B * , we obtain kM ′ 2 ([I 1 , . .…”