Timeline for Convex bodies have more volume on the outside near the boundary
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2018 at 2:59 | comment | added | Victor Protsak | It relies on two properties. Firstly, the outer prisms do not overlap and are contained in the outer $\epsilon$-neighborhood of $\partial P$. Secondly, the union of the inner prisms covers the inner $\epsilon$-neighborhood of $\partial P$. | |
| Jan 19, 2018 at 2:45 | comment | added | Victor Protsak | Hi Harry, you are right that if the projection of $O$ on $H$ falls outside $F$ then this argument does not work, even in 2D (it is easy to construct a convex polygon where any interior point $O$ projects outside at least one side=face). For these $O$ the outer containment likewise fails. However, it appears to me that Wlodek Kuperberg's modification using the prisms on faces directly (i.e., without intersecting them with cones) does the job. | |
| Jan 12, 2018 at 0:03 | comment | added | Harry Crimmins | Hi Victor, I have revisited your answer and I am not sure that is correct under all circumstances. In particular, if for some facet $F$ the origin $O$ does not lie in the set $F + \mathbb{R} n_F$, where $n_F$ is the normal vector to $F$, then I don't believe it is true that $B_\epsilon(F) \cap C^{-}$ is contained in the inward facing rectangular prism of height $\epsilon$ based on $F$. I think this inclusion also fails in the case where the origin is less than $\epsilon$ away from $F$. Do you see any way to remedy the proof? | |
| Dec 18, 2017 at 23:13 | vote | accept | Harry Crimmins | ||
| Jan 16, 2018 at 6:31 | |||||
| Dec 18, 2017 at 23:13 | comment | added | Harry Crimmins | Hi Victor, I quite like you answer. Previously I had tried to compute the volume on the outside (as in Igor's answer) and then derive the inequality by comparing this quantity to the volume on the inside, but I had never been able to do so quickly. Without getting into too many details about my previous approach, I think your formulation elegantly solves this part of the problem. Thank you for your answer! | |
| Dec 18, 2017 at 20:07 | history | edited | Victor Protsak | CC BY-SA 3.0 |
cylinder-> prism, light editing, ref to W. Kuperberg's answer
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| Dec 18, 2017 at 3:10 | history | answered | Victor Protsak | CC BY-SA 3.0 |