Return to Question

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position.

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$, or on other assumed characteristics of the point set), but not on the individual points.

Edit4: I also want to avoid dependence on $d$.

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position (in a ball of radius $R$).

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$, $R$ (or on other assumed characteristics of the point set), but not on the individual points.

Edit4: I also want to avoid dependence on $d$.

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position.

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$.

Edit4: I also want to avoid dependence on $d$.

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position.

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$, or on other assumed characteristics of the point set), but not on the individual points.

Edit4: I also want to avoid dependence on $d$.

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position.

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$.

Let $(x_1,...,x_N)$ be points in $R^d$, and $\sigma=(\sigma_1,...,\sigma_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|_2$ will denote the Euclidean norm. I am looking for a good upper estimate of:

$E_{\sigma} \sup_z \| \sum_{n=1}^N \sigma_n \frac{x_n-z}{\|x_n-z\|_2} \|_2$.

So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate?

If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?

Edit: The points $x_1,...,x_N$ are assumed to be in general position.

Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.

Edit3: The domain of $z$ is allowed to depend on $N$.

Edit4: I also want to avoid dependence on $d$.