On \(L_{p}\)-affine surface area and curvature measures. (English) Zbl 1343.52006
A new definition of \(L_p\)-affine surface area (Definition 1.5, Theorem 1.6, Definition 3.1) is given, it depends only on curvature measures. Its equivalence to older definitions is established. Important properties, such as upper semi-continuity and affine isoperimetric inequality are proven somewhat easier than for other definitions. A good survey of this area of research is given in the introduction section of this paper.
MSC:
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
| 28A75 | Length, area, volume, other geometric measure theory |
| 51M16 | Inequalities and extremum problems in real or complex geometry |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 53A15 | Affine differential geometry |
