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Совместный общематематический семинар СПбГУ и Пекинского Университета
16 декабря 2021 г. 16:00–17:00, г. Санкт-Петербург, online
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Random section and random simplex inequality
D. Zaporozhets |
Количество просмотров: |
Эта страница: | 129 |
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Аннотация:
Consider some convex body $K \subset \mathbb{R}^d$. Let $X_1,\dots,X_k$, where $k \le d$, be random points
independently and uniformly chosen in $K$, and let $\xi_k$ be a uniformly distributed
random linear $k$ - plane. We show that for $p\ge-d+k+1$,
$$ \mathbb{E} |K \cap \xi_k |^{d+p} \le c_{d,k,p} \cdot |K |^k \mathbb{E} | \mathrm{conv}(0,X_1, ..., X_k)|^p ,$$
where $|\cdot|$ and $\mathrm{conv}$ denote the volume of correspondent dimension and the convex
hull. The constant $c_{d,k,p}$ is such that for $k>1$ the equality holds if and only if $K$ is an
ellipsoid centered at the origin, and for $k=1$ the inequality turns to equality.
If $p=0$, then the inequality reduces to the Busemann intersection inequality, and if $k=d$
— to the Busemann random simplex inequality.
We also present an affine version of this inequality which similarly generalizes the
Schneider inequality and the Blaschke-Grömer inequality.
Based on a joint work with Alexander Litvak.
Язык доклада: английский
Website:
https://us02web.zoom.us/j/85667786331?pwd=YzZCU3czVHR1YjNCaGJabzJWaUxVZz09
* Meeting ID: 856 6778 6331 Password: 903126 |
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