Timeline for On Kalai's $3^{d}$ conjecture
Current License: CC BY-SA 4.0
11 events
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| Aug 5, 2021 at 12:41 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Aug 4, 2021 at 22:46 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Aug 4, 2021 at 21:32 | comment | added | Sylvain JULIEN | I like this! Looks like an homomorphism of monoids, and I'm deeply convinced that algebraic structures are often powerful enough to lead to real progress to open problems through an appropriate reformulation thereof. | |
| Aug 4, 2021 at 21:22 | comment | added | Pietro Majer | Dear Sylvain, if you attach to each polytope $A$ the “counting faces polynomial” $P_A$ whose k-th degree coefficient is the number of k-dimensional "faces" then $P_{A\times B}=P_AP_B$ | |
| Aug 4, 2021 at 20:56 | history | edited | Sylvain JULIEN | CC BY-SA 4.0 |
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| Feb 23, 2018 at 9:20 | comment | added | Sylvain JULIEN | It seems that for a $ d $-dimensional parallelotope the number of $ k $ -dimensional "faces" is given by the integer $ \binom{d}{k}2^{d-k}1^k $ appearing in the development of $ (2+1)^{d} $, while the number of $ k $-dimensional "faces"of its polar is given by $ \binom{d}{k}1^{d-k}2^k $ appearing in the development of $ (1+2)^{d} $. I guess the 2 stands for a pair of symmetric points and 1 for the center of symmetry. Has this been proven ? | |
| Aug 25, 2017 at 10:05 | comment | added | Sylvain JULIEN | Dear Gil, there's a little ambiguity in the notion of polytope. In your original conjecture, is a polytope supposed to be convex? If so, I think it should not be too hard to prove that any polytope can be obtained from a right parallelotope applying CS cuts (maybe "truncations" should be preferred, here the issue I face is purely linguistic). As CS cuts can only increase the total number of "faces" (this is to be proven rigorously but shouldn't be too much of an issue), your conjectured lower bound would be recovered rather easily. | |
| Aug 25, 2017 at 5:58 | comment | added | Gil Kalai | Dear Sylvain. one intruiging aspect of the conjecture is that equality holds for all "Hammer polytopes" namely polytopes obtained from an interval by applying a sequence of operations of two types: 1) Cartesian product 2) Moving from a polytope to its polar So testing your approach on Hammer polytopes would be a first step to try. | |
| Aug 24, 2017 at 19:37 | history | edited | Sylvain JULIEN | CC BY-SA 3.0 |
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| Aug 24, 2017 at 19:18 | comment | added | Sylvain JULIEN | More exactly, the idea is to show that a $d$-dimensional polytope with central symmetry can be deformed into a convex polytope having $(\mathbb{Z}/2)^{d}$ as a subgroup of its isometry group, as $3^{d}$ is a lower bound. | |
| Aug 24, 2017 at 19:00 | history | asked | Sylvain JULIEN | CC BY-SA 3.0 |