I recently posted this question on Math Stack Exchange, in which I asked for a published reference containing the proof of a certain fact. What I got instead was a very rough outline of a proof. I've posted bounties of 200 and 400 points and still not gotten an answer to the reference request.

At this point, I would like to ask if anybody on Math Overflow can point me to a source that contains either the required proof or something close to it.

Would this be considered an acceptable question here?


Edit. I've been asked for the sources of these definitions, and why the ones I gave were equivalent to the ones I read.

The source with a definition equivalent to (1) is Enciklopedija elementarnoj matematiki, Tom 5, which on p. 67 defines a polyhedral solid to be any set that can be divided into a finite collection of tetrahedra whose interiors do not overlap. This restriction is not essential, as the same source states that a finite union of polyhedral solids is a polyhedral solid. (And in any case, I think this isn't terribly difficult to prove.)

Definition (2), in essence, comes from Elementarnaja Geometrija by Pogorelov. A solid is defined on page 164 as the closure of an open set path-connected by polygonal lines (which is equivalent to the interior being connected). A polyhedron is defined as a solid whose boundary consists of a finite number of polygons. Polygons are apparently to be understood to mean convex polygonal plane regions here. Admittedly, I have changed this definition by eliminating the connectedness requirement and requiring the solid to be bounded. If there is to be any hope of the definition becoming equivalent to (1), these changes are necessary in order to exclude from consideration, for instance, the exterior of a tetrahedron or an infinite trihedral region, in which case the "polygons" bounding it are themselves unbounded.

What I am really interested in is any equivalence of (1) with a definition in the spirit of (2).

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    $\begingroup$ The question is vague. 'I have encountered the following two definitions of "polyhedral region" (or more accurately, definitions equivalent to these).' Where? And how are we sure they are actually equivalent to what you encountered? $\endgroup$ – user9072 Jan 22 '16 at 13:09
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    $\begingroup$ I've edited the question. I will point out in my defence that the mathematical content of the question is not vague, and the two answers I received both state that (1) and (2) are indeed equivalent. If those answers are correct, then any vagueness regarding the sources is a moot point because the mathematical premise of my question is correct. $\endgroup$ – David Jan 22 '16 at 16:56
  • $\begingroup$ I've edited my Stack Exchange question to eliminate the preface, which is not really needed. The motivation for the question is self-evident: the goal is to characterize sets satisfying (1) via a description of their boundary. $\endgroup$ – David Jan 22 '16 at 17:45
  • $\begingroup$ Thanks for the edit. You say "[t]he motivation for the question is self-evident" but which question are you now talking about? I'd agree that the mathematical question whether the two definitions are equivalent does not need much motivation and does not really need sources either (though giving them is a plus, in my opinion).. However you make a point of this mathematical question, not being the question you intend to ask. Indeed, this mathematical question seems to be answered already. Thus, in my mind the question you intend to ask here could use some motivation. $\endgroup$ – user9072 Jan 22 '16 at 19:34
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    $\begingroup$ Why/what for do you need the reference? Possible answers include: a) I want to quote it in a paper. b) I do not understand/trust the answers on Mathematics and want a more detailed/authorative one. c) I though about writing this up myself in detail, but I will not if there exists a good write up already. All of them are legitimated reasons, but they are not the same (in relevant ways). $\endgroup$ – user9072 Jan 22 '16 at 19:34
  • $\begingroup$ Finally, to answer you original meta question more directly: a question asking Def 1 in Source A and Def 2 in Source B are/seem equivalent. Is there a Source C that establishes this or related equivalence in detail. Seems like a plausibly on-topic question. (This is not my type of math though, so this is not a definite answer.) $\endgroup$ – user9072 Jan 22 '16 at 19:34
  • $\begingroup$ Quid, I meant the motivation for the mathematical question was self-evident without the sources. I didn't want my question to remain subject to criticism that it was vague. The reason I would like to have this information is closest to (b). I would like to know the proof of this fact in detail. The answers I was given, and I believe any answers to the question on Stack Exchange or Math Overflow, will necessarily omit so many details that they become difficult to verify, for me at least. I am interested in knowing the lemmas and intermediate steps that would be used in a more ordered exposition $\endgroup$ – David Jan 23 '16 at 1:29
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    $\begingroup$ Also, while I understand that you are curious about my motivation, I wonder why all of this is relevant to determining whether the question is on-topic or not. I don't agree that my motivation ought to make a big difference in terms of the kinds of answers I receive, which should be references to any literature discussing my question or closely related topics. $\endgroup$ – David Jan 23 '16 at 1:32
  • $\begingroup$ I will add that even if a published proof is similar to the one in the answer in terms of level of detail, I would still prefer it. My inclination is to trust things that have been published more than answers on MSE, and if there are gaps, I will be more motivated to try to fill them in knowing that the intermediate statements themselves are reliable because they have been made by an expert in that field. This isn't intended as a criticism of MSE or Math Overflow, but I think this is inherent in the nature of a Q&A site. $\endgroup$ – David Jan 23 '16 at 1:41
  • $\begingroup$ I recommend you ways to improve your question. The motivation that is given and the style in which the question is posed (with or without source for example) does usually affect how the question is received, viz. its on-topicnes. For example, would you just post the two statements and asked for a reference to a detailed proof, chances are somebody might take it for homework, and react negatively. $\endgroup$ – user9072 Jan 23 '16 at 2:14

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