On the deterministic interior body of random polytopes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918128284336128 |
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| author | Pafis, Minas Tziotziou, Natalia |
| author_facet | Pafis, Minas Tziotziou, Natalia |
| contents | Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We show that for any $β\in (0,1)$ there exists a constant $c(β)>0$ such that the following holds true: If $μ$ is a Borel probability measure on ${\mathbb R}^n$ then, for all $N\geq c(β)n$ we have that $K_N\supseteq T_{β\ln(\frac{N}{n})}(μ)$ with probability greater than $1-\exp(-\tfrac{1}{2}N^{1-β}n^β)$, where $T_p(μ)$ is the convex set of all points $x\in\mathbb{R}^n$ with half-space depth greater than or equal to $e^{-p}$. Our approach does not require any additional assumptions about the measure $μ$ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family $\{T_p(μ)\}_{p>0}$ to other natural families of convex bodies associated with $μ$, such as the $L_p$-centroid bodies of $μ$ or the level sets of the Cramér transform of $μ$, and use this information in order to estimate the size of a random $K_N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the deterministic interior body of random polytopes Pafis, Minas Tziotziou, Natalia Metric Geometry Probability Primary 60D05, Secondary 52A22, 52A23, 60E15, 62H05 Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We show that for any $β\in (0,1)$ there exists a constant $c(β)>0$ such that the following holds true: If $μ$ is a Borel probability measure on ${\mathbb R}^n$ then, for all $N\geq c(β)n$ we have that $K_N\supseteq T_{β\ln(\frac{N}{n})}(μ)$ with probability greater than $1-\exp(-\tfrac{1}{2}N^{1-β}n^β)$, where $T_p(μ)$ is the convex set of all points $x\in\mathbb{R}^n$ with half-space depth greater than or equal to $e^{-p}$. Our approach does not require any additional assumptions about the measure $μ$ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family $\{T_p(μ)\}_{p>0}$ to other natural families of convex bodies associated with $μ$, such as the $L_p$-centroid bodies of $μ$ or the level sets of the Cramér transform of $μ$, and use this information in order to estimate the size of a random $K_N$. |
| title | On the deterministic interior body of random polytopes |
| topic | Metric Geometry Probability Primary 60D05, Secondary 52A22, 52A23, 60E15, 62H05 |
| url | https://arxiv.org/abs/2508.15261 |