Random Inscribing Polytopes

For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem

Lokales Verhalten konvexer Körper und Approximation

Fahnenmaße konvexer Körper werden als Projektionsmittel eingeführt und Eigenschaften abgeleitet. Für gemischte Maße der translativen Integralgeometrie und für Stützfunktionen werden Darstellungssätze mittels Fahnen-Oberflächenmaßen bewiesen. Zudem wird die Approximation konvexer Körper durch zufällige Polytope und zufällige polyedrische Mengen behandelt. Es werden asymptotische Mittelwertformeln und Varianzabschätzungen bewiesen, die den Fehler bei dieser Approximation beschreiben

Random Inscribing Polytopes

International audienceFor convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem

RANDOM INSCRIBING POLYTOPES

Abstract. For convex bodies K with C 2 boundary in R d, we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem. 1