Some remarks on the geometry of convex sets. (English) Zbl 0651.52010
Geometric aspects of functional analysis, Isr. Semin. 1986-87, Lect. Notes Math. 1317, 224-231 (1988).
[For the entire collection see Zbl 0638.00019.]
The author presents a strengthening of Santalo’s inequality for the unit balls of normed spaces with 1-unconditional bases. He also proves that if \(Q_ n\) is the central unit cube in \({\mathbb{R}}^ n\) (n\(\geq 10)\) and \(C_ n\) is the central Euclidean ball of volume 1, then \(| H\cap C_ n| \geq | H\cap Q_ n|\) for every 1-codimensional subspace H of \({\mathbb{R}}^ n\).
The author presents a strengthening of Santalo’s inequality for the unit balls of normed spaces with 1-unconditional bases. He also proves that if \(Q_ n\) is the central unit cube in \({\mathbb{R}}^ n\) (n\(\geq 10)\) and \(C_ n\) is the central Euclidean ball of volume 1, then \(| H\cap C_ n| \geq | H\cap Q_ n|\) for every 1-codimensional subspace H of \({\mathbb{R}}^ n\).
Reviewer: K.Nikodem
MSC:
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
