I was thinking that there should exist a definition for "convexity" of spatial polygons. A planar convex quadrilateral that has one vertex moved (perpendicularly) out of the plane should continue to be convex in such a definition. Does someone know something in this direction?
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1$\begingroup$ What do you expect to do with such a definition? What do you need it for? $\endgroup$– Johannes HahnOct 9, 2019 at 14:10
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$\begingroup$ I do not see why this question has been put on hold. I think that what is being asked is that how can one define a notion of convexity for a space curve. This is a natural question which others might wonder about as well. elwyn even provides an example to clarify what is being asked. $\endgroup$– Mohammad GhomiOct 10, 2019 at 18:44
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A natural definition for convex space curves is given by requiring that the curve lies on the boundary of its convex hull. In this sense, the curves that you describe will be convex.
Convex space curves have been well studied and have a number of interesting properties. For instance, when they are smooth, their torsion must vanish at least four times, which is a theorem of Sedykh. See this paper for a recent generalization of this result based on Arnold's tennis balls theorem, and other references in this area.
answered Oct 9, 2019 at 14:47