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Let $D$ be an open convex subset of $\mathbf R^d$ with boundary $\partial D$ and closure $\overline D$. For $x\in\mathbf R^d$ and a unit vector $e$ let $L_{x,e}=\{x+\lambda e, \lambda\ge0\}$ be the ray drawn from $x$ in the direction $e$. For $x\in\mathbf R^d$, let $$D(x)=\{y\in \mathbf R^d \colon L_{x,y/|y|} \cap \overline D \ne \emptyset\}.$$ In particular, $D(x)=\overline D$ for all $x\in D$. Furthermore, for $y\in D(x)$, let $$\lambda(x,y/|y|)=\max\{R>0 \colon x+Ry/|y| \in \overline D\}.$$

(From Vassili Kolokoltsov: On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations, arXiv:1501.03925, DOI: 10.1515/fca-2015-0060.

enter image description here

Look at the above picture, let us consider a simple case if $d=2$. How to show $D(x)=\bar{D}, x\in D$? This results seems a little bit strange. If $x\in D$, then $L_{x,y/{|y|}}\cap \bar{D}\neq \varnothing$ forever. So $D(x)=R^2,$ why $D(x)=\bar{D}, x\in D$? Is there some misunderstanding? Please correct it.

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    $\begingroup$ Could you say where the quoted text is from: which textbook or article? $\endgroup$ Commented Jun 18, 2021 at 19:37
  • $\begingroup$ this article is "On Fully Mixed and Multidimensional Extensions of the Caputo and Riemann-Liouville Derivatives, Related Markov Processes and Fractional Differential Equations" On page 1049. $\endgroup$
    – Ailiy Evan
    Commented Jun 19, 2021 at 11:17
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    $\begingroup$ Thanks to @MartinSleziak for editing in the text from the picture, and including the reference! @‍AiliyEvans, for future reference, that is the appropriate way to proceed if a significant part of a post can only be understood from reference to an image. $\endgroup$
    – LSpice
    Commented Jun 20, 2021 at 14:21
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    $\begingroup$ Could some of the close voters comment as to why? On the face of it, this seems an okay question (now improved due to edit): it is asking about a research paper which, indeed, does seem to contain an error. My understanding of the definition is that $D(x)$ is the entire unit sphere in $\mathbb R^n$, if $x\in D$, because $D$ is open. I guess if I knew more about the area, I could hazard a guess as to what was meant, but I don't. $\endgroup$ Commented Jun 20, 2021 at 15:17
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    $\begingroup$ I concur that you should simply ask the author about this, but disagree that this means that the question should be closed - questions asking how to justify a claim in a mathematical text where the question-asker fears they may have misread are relatively common on MO, and people are often able to guess what was meant in cases the text was unclear. $\endgroup$
    – Will Sawin
    Commented Jun 21, 2021 at 18:16

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