Remarks about mixed discriminants and volumes. (English) Zbl 1292.52004
Certain inequalities for mixed discriminants of positive semi-definite matrices and mixed volumes of compact convex sets in \(\mathbb R^n\) are put forward. The authors discuss how the latter are related to the monotonicity of the information functional on the class of convex bodies, which is a geometric (dual) analogue of the classical Fisher information. In dimension 2 it is shown that the information functional is monotone with respect to the Minkowski addition. The work is partially motivated by a result of D. Hug and R. Schneider [Adv. Math. 228, No. 5, 2634–2646 (2011; Zbl 1230.52021)] regarding a certain inequality for mixed volumes of zonoids.
Reviewer: Alexey Alimov (Moskva)
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |
| 52A39 | Mixed volumes and related topics in convex geometry |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
Keywords:
mixed discriminant; positive semi-definite matrix; mixed volume; convex body; zonoid; monotonicity of informationCitations:
Zbl 1230.52021References:
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