Timeline for When is the convex hull of two space curves the union of lines?
Current License: CC BY-SA 3.0
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| Jun 11 at 15:01 | comment | added | Paata Ivanishvili | @MatteoRaffaelli, 2. Take endpoint A_1 of the space curve and start moving it towards A_2 in a continuous way ( call these points A_t, where t belongs to the interval [1,2]), and look what happens with the chords joining A_t with another point B_t of the space curve so that the chord [A_t,B_t] intersects the vertical line L. Clearly O_1=P_1, and O_2=P_2. By continuity there exists t in [1,2] such that O_t = P. | |
| Jun 11 at 15:01 | comment | added | Paata Ivanishvili | @MatteoRaffaelli, 1. Let P be any point in the interior of the convex hull V. Draw a vertical line L passing through P. It will cross the boundary of V at two points, say P_1 (upper one) and P_2 (lower one). We already know that P_1 belongs to some chord L_1 of the space curve (by chord I mean line joining two points of the space curve). And the same is true for P_2 with some another chord L_2. Let us say that L_1 has endpoints [A_1, B_1], and L_2 has endpoints [A_2,B_2]. | |
| Jun 10 at 8:04 | comment | added | Matteo Raffaelli | Would you mind providing some hints on how to show that if the boundary of the convex hull is a union of line segments, then the same holds for the interior? Thanks. | |
| Jun 15, 2015 at 12:51 | history | edited | Paata Ivanishvili | CC BY-SA 3.0 |
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| Jun 15, 2015 at 12:45 | history | edited | Paata Ivanishvili | CC BY-SA 3.0 |
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| Jun 14, 2015 at 22:12 | comment | added | Paata Ivanishvili | Boundary of convex hull is convex envelope of its extreme pints, which (mostly but of course not always) is a surface with zero gaussian curvature and hence homogeneous Monge--Amp`ere equation $\det(Hess B)=0$ and here is Bellman function. And vice versa: some extremal problems correspond to finding minimal concave functions over an obstacle, which appears to be concave envelope and hence we are talking about convex hulls. | |
| Jun 14, 2015 at 21:21 | comment | added | Joseph O'Rourke | One would never guess from the title of your paper that it bears on this problem! | |
| Jun 14, 2015 at 21:11 | comment | added | Joseph O'Rourke | The 1st-cited "this paper": Ivanisvili, Paata, Dmitriy M. Stolyarov, and Pavel B. Zatitskiy. "Bellman VS Beurling: sharp estimates of uniform convexity for $L^p$ spaces." arXiv preprint arXiv:1405.6229 (2014). (Abstract link.) | |
| Jun 14, 2015 at 21:04 | history | answered | Paata Ivanishvili | CC BY-SA 3.0 |