Preprints (rote Reihe) des Fachbereich Mathematik
Refine
Year of publication
Keywords
- average density (3)
- tangent measure distributions (3)
- Brownian motion (2)
- Palm distributions (2)
- average densities (2)
- density distribution (2)
- lacunarity distribution (2)
- occupation measure (2)
- order-two densities (2)
- Algebraic Geometry (1)
- Cantor sets (1)
- Complexity (1)
- Complexity and performance of numerical algorithms (1)
- Dirichlet series (1)
- Function of bounded variation (1)
- Hochschild homology (1)
- Hochschild-Homologie (1)
- Homologietheorie (1)
- Ill-Posed Problems (1)
- Improperly posed problems (1)
- Integral transform (1)
- Kallianpur-Robbins law (1)
- Linear Integral Equations (1)
- Local completeness (1)
- Moduli Spaces (1)
- Palm distribution (1)
- Quasi-identities (1)
- Rectifiability (1)
- Riemann-Siegel formula (1)
- Sheaves (1)
- Stratifaltigkeiten (1)
- Translation planes (1)
- Verschlüsselung (1)
- Vigenere (1)
- Zyklische Homologie (1)
- algebraic geometry (1)
- cusp forms (1)
- cyclic homology (1)
- fractals (1)
- geometric measure theory (1)
- geometry of measures (1)
- higher order (1)
- hyper-quasi-identities (1)
- hyperquasivarieties (1)
- intersection local time (1)
- invariant theory (1)
- limit models (1)
- locally maximal clone (1)
- log averaging methods (1)
- logarithmic average (1)
- logarithmic averages (1)
- moduli spaces (1)
- non-commutative geometry (1)
- order-three density (1)
- order-two density (1)
- ovoids (1)
- planar Brownian motion (1)
- preservation of relations (1)
- quadratic forms (1)
- quasivarieties (1)
- ratio ergodic theorem (1)
- singular spaces (1)
- singuläre Räume (1)
- strong theorems (1)
Faculty / Organisational entity
276
Let \(a_1,\dots,a_n\) be independent random points in \(\mathbb{R}^d\) spherically symmetrically but not necessarily identically distributed. Let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_n\) and for any \(k\)-dimensional subspace \(L\subseteq \mathbb{R}^d\) let \(Vol_L(X) :=\lambda_k(L\cap X)\) be the volume of \(X\cap L\) with respect to the \(k\)-dimensional Lebesgue measure \(\lambda_k, k=1,\dots,d\). Furthermore, let \(F^{(i)}\)(t):= \(\bf{Pr}\) \(\)(\(\Vert a_i \|_2\leq t\)),
\(t \in \mathbb{R}^+_0\) , be the radial distribution function of \(a_i\). We prove that the expectation
functional \(\Phi_L\)(\(F^{(1)}, F^{(2)},\dots, F^{(n)})\) := \(E(Vol_L(X)\)) is strictly decreasing in
each argument, i.e. if \(F^{(i)}(t) \le G^{(i)}(t)t\), \(t \in {R}^+_0\), but \(F^{(i)} \not\equiv G^{(i)}\), we show \(\Phi\) \((\dots, F^{(i)}, \dots\)) > \(\Phi(\dots,G^{(i)},\dots\)). The proof is clone in the more general framework
of continuous and \(f\)- additive polytope functionals.