Timeline for Concavity of mixed volumes and mixed discriminants
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9 events
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| May 25, 2015 at 19:25 | comment | added | Suvrit | Ok, that clarifies my doubts. Before plowing into this, however, I'd like to ask you do you have any numerical evidence that suggests that your conjecture may hold? Thanks! | |
| May 25, 2015 at 18:01 | comment | added | Sasho Nikolov | For example, the simplest $i=j=1$ case of my question reduced to the inequality $D(A, B', C_3, \ldots, C_n)/2 + D(A',B, C_3, \ldots, C_n)/2 \ge \sqrt{D(A, B, C_3, \ldots, C_n)D(A',B',C_3, \ldots, C_n)}$. I dont see how to connect this to concavity of the mixed discriminant on the discrete simplex. | |
| May 25, 2015 at 17:55 | comment | added | Sasho Nikolov | @Suvrit Gardner's Bulletin paper is indeed a great resource. The other paper you link to talks about a different concavity question that goes back to Gromov. They fix $n$ bodies and look at mixed volume as a function on the discrete simplex $\{x \in \mathbb{Z}_+^n: \sum x_i = n\}$, and ask if the function is log concave. Aleksandrov-Fenchel says that it is log-concave on particular segments, and this turns out to be equivalent to generalized Brunn-Minkowski. However, I don't see a similar connection between the concavity of my function $h$ and concavity on the discrete simplex. | |
| May 25, 2015 at 14:38 | comment | added | Suvrit | The closest paper I found relevant to your question is: arxiv.org/pdf/1412.8200.pdf --- does Corollary 4.1 in there help? | |
| May 25, 2015 at 14:27 | comment | added | Suvrit | The discussion on pg. 391 of ams.org/journals/bull/2002-39-03/S0273-0979-02-00941-2/… seems to contain useful pointers; the conjecture seems nice but probably it breaks. | |
| May 25, 2015 at 9:11 | history | edited | Sasho Nikolov | CC BY-SA 3.0 |
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| May 25, 2015 at 5:19 | history | edited | Sasho Nikolov | CC BY-SA 3.0 |
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| May 25, 2015 at 2:07 | history | edited | Sasho Nikolov | CC BY-SA 3.0 |
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| May 25, 2015 at 1:52 | history | asked | Sasho Nikolov | CC BY-SA 3.0 |