scholarly journals Bézout-Type Inequality in Convex Geometry

2017 ◽  
Vol 2019 (16) ◽  
pp. 4950-4965 ◽  
Author(s):  
Jian Xiao
Keyword(s):  
Complex Geometry ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Type Inequality ◽  
Mixed Volumes

Abstract Inspired by a result of Soprunov and Zvavitch, we present a Bézout type inequality for mixed volumes, which holds true for any convex bodies and improves the previous result. The key ingredient is the reverse Khovanskii–Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.

scholarly journals Correspondences between convex geometry and complex geometry

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Brian Lehmann ◽  
Jian Xiao
Keyword(s):  
Complex Geometry ◽  
Convex Geometry ◽  
Line Bundles ◽  
Mixed Volumes ◽  
Projective Varieties ◽  
Body Construction ◽  
Cone Of Curves ◽  
Smooth Projective Varieties ◽  
Holomorphic Line Bundles ◽  
Effective Cone

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves. Comment: EpiGA Volume 1 (2017), Article Nr. 6

Determination of Boolean models by densities of mixed volumes

2019 ◽  
Vol 51 (01) ◽  
pp. 116-135
Author(s):  
Daniel Hug ◽  
Wolfgang Weil
Keyword(s):  
Uniqueness Theorem ◽  
Representation Theorem ◽  
Convex Bodies ◽  
Boolean Models ◽  
Mixed Volumes ◽  
Shape Information ◽  
Translation Invariant ◽  
Mean Values ◽  
Particle Process

AbstractIn Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

scholarly journals REMARKS ABOUT MIXED DISCRIMINANTS AND VOLUMES

2014 ◽  
Vol 16 (02) ◽  
pp. 1350031 ◽  
Author(s):  
S. ARTSTEIN-AVIDAN ◽  
D. FLORENTIN ◽  
Y. OSTROVER
Keyword(s):  
Fisher Information ◽  
Convex Sets ◽  
Compact Convex ◽  
Convex Bodies ◽  
Mixed Volumes ◽  
Information Functional ◽  
Geometric Analogue

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in ℝn. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

Section means, integral transforms, and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 332-333
Author(s):  
Paul Goodey ◽  
Markus Kiderlen ◽  
Wolfgang Well
Keyword(s):  
Surface Area ◽  
Support Function ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Integral Transforms ◽  
Boolean Models ◽  
Area Measure ◽  
Surface Area Measure ◽  
Left Hand ◽  
The Mean

For a stationary particle process X with convex particles in ℝdd ≧ 2, a mean body M(X) can be defined by where h(M,·) denotes the support function of the convex body M, γ the intensity of X, and P0 is the distribution of the typical particle of X (a probability measure on the set of convex bodies with Steiner point at the origin). Replacing the support function h(M,·) by the surface area measure S(M,·) (see Schneider (1993), for the basic notions from convex geometry), we get the Blaschke body B(X) of X, After normalization, the left-hand side represents the mean normal distribution of X. The main problem discussed here is whether B(X) (respectively S(B(X), ·)) is uniquely determined by the mean bodies M(X ∩ E) in random planar sections X ∩ E of X. From more general results in Weil (1995), it follows that the expectation ES(M(X ∩ E), ·) (taken w.r.t. the uniform distribution of two-dimensional subspaces E in ℝd) equals the surface area measure of a section mean B2(B(X)) of B(X). Thus, the formulated stereological question can be reduced to the injectivity of the transform B2 : K ↦ B2(K).

Algebraically inspired results on convex functions and bodies

2016 ◽  
Vol 18 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Liran Rotem
Keyword(s):  
Convex Functions ◽  
Final Section ◽  
Polar Body ◽  
Convex Bodies ◽  
Type Inequality ◽  
Dual Function ◽  
Minkowski Addition ◽  
Geometric Means ◽  
Better Than

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function [Formula: see text] one has [Formula: see text] where [Formula: see text]. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.

scholarly journals Flag representations of mixed volumes and mixed functionals of convex bodies

2018 ◽  
Vol 460 (2) ◽  
pp. 745-776 ◽  
Author(s):  
Daniel Hug ◽  
Jan Rataj ◽  
Wolfgang Weil
Keyword(s):  
Convex Bodies ◽  
Mixed Volumes
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