Időpont: 2025. július 1. 12:30

Helyszín: Riesz-terem, Bolyai Épület, 6720 Szeged, Aradi vértanúk tere 1.

Előadó: Papvári Dániel (Szegedi Tudományegyetem)

Előadás címe: Véletlen politópok súlyozott vegyes térfogatainak várható értéke

Absztrakt: Let $K$ be a convex body in $\mathbb R^d$, let $j\in \{1,\ldots,d-1\}$ and let $\varrho$ be a suitable probability density function with respect to the $d$-dimensional Hausdorff measure on $K$. Denote by $K_{(n)}$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the probability distribution determined by $\varrho$.

For the case when $\varrho\equiv 1/V(K)$ and $\partial K$ is $C^2_+$, Reitzner (2004) proved an asymptotic formula for the expectation of the difference of the $j$th intrinsic volumes of $K$ and $K_{(n)}$, as $n\to\infty$. B\""or\""oczky, Hoffmann, and Hug (2008) extended this result to the case when $\varrho\equiv 1/V(K)$ and the only condition on $K$ is that a ball rolls freely in $K$. B\""or\""oczky, Fodor, Reitzner, and V\'igh (2009) also showed that in general, the assumption of the existence of a rolling ball inside $K$, for the mean width, cannot be dropped.

B\"or\"oczky, Fodor, and Hug (2010) proved an asymptotic formula for the weighted volume approximation of $K$ under no smoothness assumptions on $\partial K$. We study the expectation of weighted intrinsic volumes for random polytopes generated by non-uniform probability distributions in convex bodies with very mild smoothness conditions.