What is the meaning of $\mathbb{R}_{++}$?

Question

From [1], on page 9, one can find the following expression.

$$L:\mathbb{R}\setminus\mathcal{A}\rightarrow\mathbb{R}_{++}$$

What is $\mathbb{R}_{++}$ is meant to be?

The $\mathbb{R}_{+}$ symbol seems to usually mean positive real numbers or non-negative real numbers, but I've never seen (and cannot find anything) about $\mathbb{R}_{++}$.

[1] - Beck, Amir and Shoham Sabach. “Weiszfeld’s Method: Old and New Results.” Journal of Optimization Theory and Applications 164 (2015): 1-40. https://doi.org/10.1007/s10957-014-0586-7

This is non-standard but my guess is $\mathbb R_{+}=[0,\infty)$ and $\mathbb R_{++}=(0,\infty)$. — Kavi Rama Murthy, Jul 09 '19 at 23:23
Shame on them for not including it in “Appendix A: Notations.” I can’t seem to access the PDF; could you pull a block quote of the whole paragraph? — gen-ℤ ready to perish, Jul 09 '19 at 23:30
another thought I had, was maybe an old notation for $\mathbb{R}^2$ on the positive quadrant ? — , Jul 09 '19 at 23:58

score 2 · Accepted Answer · answered Jul 09 '19 at 23:55

The full quote is:

Another observation, which appears in Kuhn and Kuenne [12], is that Weiszfeld’s method is, in fact, a gradient method. Indeed, a simple computation shows that an alternative representation of the operator $T$ is given by $$ \renewcommand{\vec}[1]{\mathbf{#1}} T(\vec{x}) = \vec{x} - \frac{1}{L(\vec{x})} \nabla f(\vec{x}) \qquad(\vec{x}\not\in\cal{A}), \tag{11} $$ where the operator $L:\mathbb{R}^d\setminus\cal{A} \to \mathbb{R}_{++}$ is defined by $$ L(\vec{x}) := \sum_{i=1}^{m} \frac{\omega_i}{\|\vec{x}-\vec{a}_i\|}. \tag{12} $$

On page 4 of the same paper, the authors note that $\omega_i > 0$ for all $i=1,2,\dotsc,m$, from which it follows that each summand in (12) must be positive, and so $L(x) > 0$ for any $x\not\in\cal{A}$. Given this context, it seems that the notation $\mathbb{R}_{++}$ is meant to denote the open half-line, i.e. $\mathbb{R}_{++} = (0,\infty)$. One suspects that the authors are simply trying to emphasize that $L(x)$ is strictly positive, and cannot ever be zero.

What is the meaning of $\mathbb{R}_{++}$?

1 Answers1