Minas Pafis
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View article: On the deterministic interior body of random polytopes
On the deterministic interior body of random polytopes Open
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…
View article: Half-space depth of log-concave probability measures
Half-space depth of log-concave probability measures Open
Given a probability measure $$\mu $$ on $${{\mathbb {R}}}^n$$ , Tukey’s half-space depth is defined for any $$x\in {{\mathbb {R}}}^n$$ by $$\varphi _{\mu }(x)=\inf \{\mu (H):H\in {{{\mathcal {H}}}}(x)\}$$ …
View article: Threshold for the expected measure of the convex hull of random points with independent coordinates
Threshold for the expected measure of the convex hull of random points with independent coordinates Open
Let be an even Borel probability measure on . For every , consider independent random vectors in , with independent coordinates having distribution . We establish a sharp threshold for the product measure of the random polytope in under th…
View article: Threshold for the expected measure of random polytopes
Threshold for the expected measure of random polytopes Open
Let $$\mu $$ be a log-concave probability measure on $${\mathbb R}^n$$ and for any $$N>n$$ consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ , where $$X_1,X_2,\ldots $$ are inde…
View article: Threshold for the expected measure of the convex hull of random points with independent coordinates
Threshold for the expected measure of the convex hull of random points with independent coordinates Open
Let $μ$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $μ$. We establish a s…
View article: Threshold for the expected measure of random polytopes
Threshold for the expected measure of random polytopes Open
Let $μ$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed accord…
View article: Half-space depth of log-concave probability measures
Half-space depth of log-concave probability measures Open
Given a probability measure $μ$ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $φ_{μ}(x)=\inf\{μ(H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ cont…