[00:00:10.21] Thank you so will start to stop. [00:00:14.23] [00:00:16.11] And say during. [00:00:17.06] [00:00:20.22] The parallel section function so there is a direction. [00:00:25.15] [00:00:27.10] Distance d. from the origin. [00:00:29.08] [00:00:30.23] And this area is. [00:00:32.10] [00:00:34.04] A cake c. of c.. [00:00:35.20] [00:00:41.14] So this is then minus one dimensional volume of the section of k. [00:00:45.04] [00:00:45.04] by the hyper plane or the scale approach at the t.. [00:00:50.05] [00:00:57.03] Where proving the formula. [00:00:58.23] [00:01:01.02] For the fractional derivative of the order q. 0 for its course sign of q. [00:01:08.17] [00:01:08.17] over 2 by and minus 2 minus one. [00:01:13.18] [00:01:15.19] And the forward transform. [00:01:17.15] [00:01:20.07] Of the norm to the power and plus 2 plus one. [00:01:23.08] [00:01:25.10] Of the point c.. [00:01:26.12] [00:01:30.06] And stop we had an expression for [00:01:33.17] [00:01:35.01] this parallel section function with if so it's enough to consider [00:01:40.01] [00:01:40.01] fractional derivatives of the orders between negative $1.00 and [00:01:43.20] [00:01:43.20] 0 if we prove this formula it's extended and early to call it the values of q.. [00:01:49.08] [00:01:51.13] So it's a constant into go with this fear the scale of product. [00:01:59.20] [00:02:01.11] To minus one minus q. and. [00:02:03.14] [00:02:05.10] Norm that. [00:02:06.05] [00:02:08.07] To the minus and plus give plus one that so that's where we stop. [00:02:15.12] [00:02:17.06] This is fork c. in the units fear but [00:02:21.09] [00:02:21.09] 1st of all extend this function to the whole hour and there's a come out genius [00:02:25.12] [00:02:25.12] function of degree minus one minus Q Ok so now season aren't. [00:02:31.03] [00:02:34.20] Now with Day can even test function. [00:02:37.09] [00:02:42.07] By. [00:02:42.19] [00:02:44.23] The ply this function of seat to. [00:02:47.12] [00:02:51.01] See q. [00:02:53.23] [00:02:53.23] of 0 is considered as a function of c it's come a genius of degree minus one minus. [00:02:59.23] [00:03:01.17] 25 c.. [00:03:02.18] [00:03:06.02] And it's by integration so we integrate this over r n. [00:03:12.10] [00:03:29.22] We integrate this with the function 5 c. Dixie. [00:03:33.02] [00:03:35.02] So this is a and [00:03:38.13] [00:03:38.13] the action is integration now we change the order of integrations. [00:03:44.03] [00:03:45.19] The Into go over us and minus one goes outside. [00:03:48.20] [00:03:57.23] And the Into go r. and side. [00:04:00.00] [00:04:15.01] And now we are going to fight with this expression for a few minutes. [00:04:19.12] [00:04:21.03] So 1st of all the make a change variables with say that this is t. So [00:04:26.10] [00:04:26.10] this is equal to in to go of minus one minus 2 over par. [00:04:33.05] [00:04:35.05] And here because. [00:04:36.02] [00:04:40.19] There add and transform of the function. [00:04:42.17] [00:04:51.03] So that's a to get. [00:04:52.07] [00:04:53.20] And now we use 2 facts. [00:04:55.11] [00:04:57.06] First this is for a transform of z. to the power Q. [00:05:03.04] [00:05:08.10] It's constant dependent q.. [00:05:11.05] [00:05:12.15] To minus one minus Q Ok so [00:05:16.17] [00:05:16.17] this is an easy computation this is a comedian has distribution of degree q. So [00:05:21.14] [00:05:21.14] the Transformers come out genius of a degree a minus one minus q. [00:05:25.03] [00:05:25.03] on the line there is nothing else but this so [00:05:28.04] [00:05:28.04] the whole problem is how to compute the constant They don't care about constants [00:05:32.19] [00:05:34.10] and then I want to know what is the for it transform of this rather than transform so [00:05:39.18] [00:05:39.18] they consider this as a function of t.. [00:05:42.00] [00:05:44.16] And compute the transform at the point z. So this is a 50. [00:05:51.10] [00:05:55.09] F. at z. Is it a minus z. and this function. [00:06:03.17] [00:06:15.12] So this is a function f. of t. and this is the returns from of the z. [00:06:21.13] [00:06:21.13] and I would do the same thing over our story now right it does into [00:06:26.15] [00:06:26.15] the lower Our an ecstasy steeds so it's easy to minus. [00:06:32.01] [00:06:33.07] 0 c. 5 x. dx [00:06:37.19] [00:06:40.20] so we get there for a transform of the function $5.00 c.. [00:06:46.17] [00:06:56.21] So now this is therefore a transform of z. to the power q. [00:07:02.01] [00:07:04.01] apply to this function this is the same as z. to the power q. [00:07:09.09] [00:07:09.09] apply to the free transfer of this function by definition so. [00:07:15.03] [00:07:18.00] Zita the power q. is this applied to this test function [00:07:23.12] [00:07:23.12] we remove the 4 returns from and put it here so we get that expression so finally [00:07:28.19] [00:07:28.19] we get in to go over this fear norm [00:07:35.01] [00:07:38.07] the minus and plus 2 plus one and [00:07:43.19] [00:07:43.19] this becomes into go over our z. to the power q.. [00:07:50.02] [00:07:52.07] Raises. [00:07:53.00] [00:07:59.10] And the transformers that of the. [00:08:02.09] [00:08:04.19] D.c. get it. [00:08:08.08] [00:08:10.20] Ok so this is where we stop and now we look at drive the distribution. [00:08:17.22] [00:08:20.20] Bly. [00:08:21.08] [00:08:26.01] Function the ply this the distribution to the function. [00:08:29.21] [00:08:30.22] The same. [00:08:31.10] [00:08:33.01] And we are supposed to get. [00:08:34.11] [00:08:35.23] The same result because this is supposed to be equal to that with the constant So [00:08:41.21] [00:08:41.21] let's see if we do get the same result 1st of all this is. [00:08:45.12] [00:08:46.12] The norm itself by definition of the for transform. [00:08:49.23] [00:08:52.08] Applied to the function feek at. [00:08:54.08] [00:08:56.09] And this is by integration. [00:08:58.02] [00:09:10.22] And I will write this in polar coordinates x. is. [00:09:14.16] [00:09:19.08] So it's an to go over this fear Argos get nor that. [00:09:25.05] [00:09:26.23] The. [00:09:28.09] [00:09:28.09] Q. plus one and then to go from 0 to infinity are turned minus $1.00 and [00:09:35.04] [00:09:35.04] $2.00 minus plus 2 plus ones to the q.. [00:09:39.08] [00:09:42.05] The that are that are. [00:09:45.07] [00:09:49.16] And everything is even so I can write that. [00:09:53.20] [00:09:55.18] From negative infinity infinity and divide by 2. [00:09:58.16] [00:10:00.21] And this is exactly the same expression as the. [00:10:02.23] [00:10:07.13] Ok. [00:10:08.01] [00:10:09.03] So these 2 distributions as a function of c. [00:10:13.07] [00:10:15.19] act on a new test function the same same way so they're equal. [00:10:21.03] [00:10:23.16] And this is. [00:10:24.10] [00:10:26.09] The formula that we used. [00:10:27.20] [00:10:29.23] Fractional do it with if Sorry quote the usual do it is excuse an integer. [00:10:34.03] [00:10:37.01] And let me just before I start the 2nd [00:10:40.10] [00:10:41.18] topic let me just mention what if. [00:10:46.17] [00:10:52.08] If we get a cake see. [00:10:54.23] [00:10:57.01] 0 Yes the function itself a 0 which is. [00:11:01.23] [00:11:05.23] The area of the central section Ok And it's a constant which is. [00:11:13.02] [00:11:14.06] One over by an minus one and for a transform. [00:11:20.23] [00:11:24.02] Minus and plus one. [00:11:25.14] [00:11:28.02] Point c.. [00:11:29.01] [00:11:30.23] So we will use this formula later. [00:11:33.05] [00:11:34.15] But for example from this formula immediately get the phone can come ski [00:11:39.03] [00:11:39.03] theorem if all central sections of k.. [00:11:43.04] [00:11:45.03] Are equal to central sections of. [00:11:48.20] [00:11:51.10] And symmetric because this is for symmetric and the. [00:11:54.16] [00:11:57.05] Then this means that the 4 It transforms are equal so [00:12:01.05] [00:12:01.05] this becomes just the nicknames there for the 4 we transfer so [00:12:05.07] [00:12:05.07] the norms are equal Ok so this is. [00:12:09.21] [00:12:11.09] One of placation and now the 2nd topic. [00:12:13.22] [00:12:17.05] Will be. [00:12:17.19] [00:12:19.02] Slicing inequalities for functions. [00:12:21.12] [00:12:43.01] So this is completely different to are done with algebra. [00:12:45.23] [00:12:47.02] Fortunately. [00:12:47.21] [00:12:51.04] So. [00:12:52.16] [00:12:52.16] What is the problem suppose the cover star board dk. [00:12:56.17] [00:13:02.05] A star board the means. [00:13:03.20] [00:13:06.03] The origin is an interior point. [00:13:08.11] [00:13:10.19] Every straight line passing through the origin causes the boundary to exactly 2 [00:13:15.13] [00:13:16.13] points and the boundaries continuous. [00:13:20.02] [00:13:21.04] Which means that the functional of k. which is defined as. [00:13:26.12] [00:13:27.14] The norm really related to con the body but now it's not the norm. [00:13:31.06] [00:13:38.21] It's cantinas and. [00:13:44.08] [00:13:45.23] If x. is a unit vector. [00:13:47.23] [00:13:49.22] The norm of x.. [00:13:53.22] [00:13:54.23] To negative one is their ideas of k. in the direction of fix. [00:13:58.13] [00:14:00.01] So the norm is one over their ideas. [00:14:02.04] [00:14:03.12] And there is a dollar formula for William so [00:14:07.21] [00:14:07.21] well you know of Kaizen to go over this fear of ideas to the power. [00:14:14.19] [00:14:16.08] Which you're right this way this is their idea to that. [00:14:21.01] [00:14:26.22] So what is the setting so suppose we cover star body was well one. [00:14:32.11] [00:14:39.10] Bob Vila to density and k. just any measurable function. [00:14:42.20] [00:14:49.22] No negative function on k.. [00:14:55.16] [00:14:57.08] Into a goal equal to one. [00:14:58.20] [00:15:05.07] And the question is does there exist. [00:15:07.15] [00:15:13.19] Direction. [00:15:14.14] [00:15:20.20] C. so that they entered. [00:15:22.10] [00:15:25.01] Or were this section of k. by the central [00:15:30.06] [00:15:30.06] plane of the pinnacle at the direction is greater or equal to a constant. [00:15:35.23] [00:15:37.21] Which doesn't depend on k. [00:15:41.01] [00:15:41.01] And so the constant should for durak for all k. a and. [00:15:44.23] [00:15:47.12] So about. [00:15:49.04] [00:15:50.06] Something a go I would say I want to be independent defend Now I don't. [00:15:54.20] [00:16:03.15] See is not dependent. [00:16:07.08] [00:16:13.07] On k. f.. [00:16:14.11] [00:16:21.21] So. [00:16:23.07] [00:16:23.07] Every By be able to density on the set of measure one has a big enough section. [00:16:30.12] [00:16:35.15] Now go show later then this. [00:16:38.06] [00:16:40.10] Is a greater than. [00:16:42.00] [00:16:43.10] An overt square with a fan. [00:16:45.08] [00:16:50.10] And now it is known that it has to be one over the square with a friend [00:16:56.22] [00:16:56.22] but there is a very interesting dependence on. [00:17:01.03] [00:17:03.11] Of this constant on the budget k. So if case convex we do get one o. [00:17:09.12] [00:17:09.12] the school to offend so if we suppose the case convex. [00:17:14.10] [00:17:16.22] If good but [00:17:17.19] [00:17:17.19] if for example an unconditional convex boarded then this becomes [00:17:22.18] [00:17:22.18] an absolute constant It doesn't depend on the dimension it's one over to the. [00:17:27.00] [00:17:31.18] Bits the unit both of subspace will be then it's also an absolute constant [00:17:35.23] [00:17:35.23] depending on the. [00:17:37.07] [00:17:39.10] Other kind there exist very exotic. [00:17:41.09] [00:17:43.05] Specially chosen the Googles can bodies [00:17:46.08] [00:17:46.08] which discussed to be one over the square with the. [00:17:49.06] [00:17:51.20] So there is an interesting dependence and k. the constant changes dramatically. [00:17:56.02] [00:17:57.13] Dependence afaik something about. [00:18:00.16] [00:18:02.07] Is not clear it's an interesting question but the fact is there are just one. [00:18:08.09] [00:18:12.11] So let's write this problem in a different form. [00:18:15.09] [00:18:26.20] The question is does there list a constant c. that for any. [00:18:31.13] [00:18:33.06] Star. [00:18:33.18] [00:18:37.23] And f. measurable function now I don't assume the volume of [00:18:43.04] [00:18:43.04] cases one and this bill are to measure in general. [00:18:48.16] [00:18:49.17] We have this in the quote. [00:18:51.03] [00:19:07.21] So this is an equivalent formulation to what I just does here. [00:19:11.07] [00:19:14.02] Because this big this if this is one and this is one then the constant is one over [00:19:19.22] [00:19:19.22] this c. small Ok so now we are looking for. [00:19:25.14] [00:19:27.00] C. and the smallest see that works for [00:19:35.16] [00:19:35.16] all k. And so this in the current from relation. [00:19:41.02] [00:19:46.11] So if f. is identical it won [00:19:48.10] [00:19:50.10] we get volume of k. And here we get. [00:19:55.06] [00:19:57.13] There this lies and k. [00:20:01.12] [00:20:01.12] to the one over and so this is the slicing problem of will gain. [00:20:07.08] [00:20:16.06] And at the moment it is known that the constant can be improved from the square [00:20:21.00] [00:20:21.00] root defend here to the 1st suit of fan [00:20:24.19] [00:20:27.02] and it was 1st proved by both again himself [00:20:29.17] [00:20:29.17] with the next door that there is mid-term and the very mature was removed. [00:20:33.04] [00:20:35.04] So just by putting Affleck will do one you don't make [00:20:39.05] [00:20:39.05] such a dramatic progress move from the defend to the through the. [00:20:43.23] [00:20:46.17] Fence and consider. [00:20:47.22] [00:20:50.11] But what about a cluster of functions is the middle for [00:20:54.20] [00:20:54.20] example decreasing densities and so on this is an open question So [00:20:58.19] [00:20:58.19] what happens there is a pretty good control [00:21:03.08] [00:21:03.08] over this problem with respect to the body but not the function. [00:21:07.19] [00:21:16.00] So what I'm going to prove. [00:21:17.16] [00:21:19.22] Today it girl in the loads me. [00:21:24.21] [00:21:28.17] I look up to stuff but. [00:21:29.22] [00:21:31.03] I. [00:21:32.06] [00:21:32.06] Don't know. [00:21:33.20] [00:21:41.04] So I will prove that star boy. [00:21:43.17] [00:21:53.19] And. [00:21:54.07] [00:21:56.01] Any measurable function f.. [00:21:57.21] [00:22:03.16] Then to go. [00:22:04.13] [00:22:07.21] Over a case smaller than. [00:22:09.13] [00:22:10.19] Do the outer William ratio distance from Kate to the class of interest section. [00:22:16.18] [00:22:23.06] And then the same thing as before. [00:22:25.11] [00:22:34.00] So this works as a constant c. So what is out there William ratio distance. [00:22:41.01] [00:22:55.17] This is the. [00:22:56.11] [00:22:58.22] So we have k. which doesn't belong to the class offender section but [00:23:04.19] [00:23:04.19] is and well looking for d that includes. [00:23:08.02] [00:23:09.19] And has the smallest possible. [00:23:11.13] [00:23:14.09] So this is. [00:23:15.07] [00:23:23.07] Of the Orwellian Kate to the. [00:23:26.08] [00:23:29.19] K's inside of d. and d. is an intersection bogeyman. [00:23:34.06] [00:23:38.16] Ok Now the question is what is an intersection book. [00:23:42.06] [00:23:45.12] And this is something the going to do for a little. [00:23:48.01] [00:23:51.09] To give the definitions takes. [00:23:53.08] [00:23:56.23] So the class offender section budget was introduced by Lew took. [00:24:00.23] [00:24:07.11] In 1988. [00:24:08.23] [00:24:12.12] And. [00:24:12.23] [00:24:14.22] It's kind of for ridiculous Jemmett to construction so it's gold star. [00:24:22.18] [00:24:28.13] It's called. [00:24:29.06] [00:24:32.04] The intersection board. [00:24:33.11] [00:24:40.23] Of another star board a k. l.. [00:24:44.18] [00:24:48.08] If their ideas of k. in every direction. [00:24:51.23] [00:24:56.09] X. is in a tricked or which is the norm of facts [00:25:02.04] [00:25:02.04] in k. to the power minus one is equal to the and [00:25:06.21] [00:25:06.21] minus one dimensional volume of the section. [00:25:09.09] [00:25:10.11] By the perpendicular hyperplane Ok so what we do is start with l.. [00:25:17.22] [00:25:21.14] We choose a direction take the central section of fell. [00:25:26.10] [00:25:28.08] We get the number this is their ideas of a new book in the direction c. [00:25:34.02] [00:25:35.21] Ok we take another direction take another section this is their ideas [00:25:41.16] [00:25:41.16] so we get their ideas in every direction to construct something. [00:25:44.23] [00:25:48.20] Now we continue this a little bit. [00:25:50.23] [00:25:52.04] Right the polar formula. [00:25:53.18] [00:25:55.02] It's run over and [00:25:56.01] [00:25:56.01] minus one because the board isn't minus one dimensional this one. [00:26:01.04] [00:26:03.23] This fear is. [00:26:05.01] [00:26:09.10] The section of fence and minus one bike see are tugging on [00:26:12.21] [00:26:17.06] and then the norm of l. to the minus and [00:26:23.08] [00:26:23.08] plus one that So this is the polar form of the follow from [00:26:28.13] [00:26:28.13] Ulip like to this edition and this is this farrago right and transform. [00:26:34.22] [00:26:39.03] Of the function norm l. to the bar minus and plus one. [00:26:43.08] [00:26:45.04] Point c. So what is this fair go read and transform. [00:26:48.23] [00:26:55.20] This farrago I don't transform as an operator from [00:26:59.08] [00:26:59.08] a continuous function on this fear to continuous functions in the sphere. [00:27:03.09] [00:27:08.22] Such that fast. [00:27:10.17] [00:27:12.11] Can you point c In this fear is in to go. [00:27:15.14] [00:27:20.19] Off half over this section of this year by the perpendicular hyperplane. [00:27:25.03] [00:27:29.20] So here the function is this one integrated slices. [00:27:37.20] [00:27:39.20] So to get this. [00:27:40.22] [00:27:46.01] So in other words. [00:27:47.19] [00:27:49.02] This is anything positive essentially because there's a Starbucks. [00:27:52.21] [00:27:56.01] So. [00:27:56.13] [00:27:58.09] An intersection is the intersection bloody office. [00:28:01.02] [00:28:03.10] Norm to the power negative one is there add and transform of a positive function [00:28:07.17] [00:28:10.04] Ok if there exists a positive function whose read and [00:28:13.18] [00:28:13.18] transform is this then we have this equality. [00:28:16.10] [00:28:22.18] Idea so a little more general. [00:28:24.16] [00:28:25.20] Milley is a measure on this year. [00:28:27.14] [00:28:31.10] What is the read and transform of a matter. [00:28:33.20] [00:28:35.22] This is a functional on the space of continuous functions that acts. [00:28:42.09] [00:28:45.17] Like new acts on their add and transform effect. [00:28:49.05] [00:28:52.00] Which is by integration. [00:28:53.12] [00:29:10.03] Ok so this is a continuous functional on the space See office and [00:29:16.12] [00:29:16.12] minus one which is also a measure so that I didn't measure is a measure. [00:29:20.17] [00:29:22.22] And we want to say that. [00:29:25.04] [00:29:27.20] This I need everything. [00:29:30.22] [00:29:36.09] Which one. [00:29:36.23] [00:29:39.19] But you can't see anything there. [00:29:41.12] [00:29:42.23] You can like Ok. [00:29:46.03] [00:29:48.01] Ok then I'll write here. [00:29:49.07] [00:29:53.01] So. [00:29:53.16] [00:29:54.16] I start body case golden intersection boarding. [00:29:57.17] [00:30:10.05] If. [00:30:10.17] [00:30:12.12] The norm of. [00:30:13.08] [00:30:14.12] The minus swine is equal to or I've been trying this is very correct and [00:30:19.15] [00:30:19.15] transform of some measure mill if there exists a measure mill as functionals. [00:30:27.16] [00:30:34.05] On the space of continuous function of this here or. [00:30:37.15] [00:30:40.12] So this is a function that acts on continuous functions on this fear by [00:30:44.15] [00:30:44.15] integration. [00:30:45.13] [00:30:53.13] And this must be equal to their I don't transform of like the same function f. [00:30:59.18] [00:30:59.18] So they're equal as functionals. [00:31:01.23] [00:31:03.15] Which is the integral over this fear of. [00:31:08.14] [00:31:09.16] The right and transform of f. against the measure mew [00:31:16.07] [00:31:17.13] So this must be true for a new continuous function. [00:31:22.07] [00:31:24.11] And lines one and this is the classified section body so [00:31:28.20] [00:31:28.20] now there is no dependence on off something [00:31:32.14] [00:31:34.02] don't say that it's in the section but if something is just an intersection. [00:31:37.23] [00:31:41.13] Ok. [00:31:42.06] [00:31:46.22] Let me show that. [00:31:48.01] [00:31:58.04] B. is an intersection. [00:32:00.05] [00:32:06.12] And is the bowl. [00:32:09.02] [00:32:29.20] And there is a very clear carnation connection with the forward transform [00:32:33.05] [00:32:33.05] immediately. [00:32:33.22] [00:32:35.05] Because if we take the function of our nowhere one c one squared [00:32:40.17] [00:32:42.09] one over one plus say and squared. [00:32:47.01] [00:32:48.13] What is its for transfer function of c.. [00:32:52.18] [00:33:01.00] Heat to the minus. [00:33:02.10] [00:33:03.18] One. [00:33:04.06] [00:33:05.08] And so. [00:33:06.06] [00:33:07.22] It's it to minus one or. [00:33:10.23] [00:33:13.22] Ok. [00:33:14.10] [00:33:17.10] Now if. [00:33:18.03] [00:33:20.21] We consider their add and [00:33:25.04] [00:33:25.04] transform of this function. [00:33:28.19] [00:33:40.16] So we integrate over. [00:33:41.23] [00:33:43.11] The Quite there are plenty of x. distance d. [00:33:47.04] [00:33:47.04] from the origin Ok then this is there have [00:33:52.04] [00:33:52.04] been times from that this started in the previous topic everybody remembers. [00:33:56.18] [00:33:58.21] So if you have. [00:33:59.23] [00:34:04.10] Such a function it's for a transfer. [00:34:06.17] [00:34:09.11] Of the t. of z. x.. [00:34:13.09] [00:34:23.19] So the for it transform of this. [00:34:26.01] [00:34:28.05] Is it to minus. [00:34:29.13] [00:34:30.19] 2 x. one Ok this apply to. [00:34:38.06] [00:34:40.06] Where is e. [00:34:41.01] [00:34:45.11] Ok. [00:34:45.23] [00:34:52.01] Ok now what if we integrate this function from negative infinity to infinity or [00:35:00.10] [00:35:00.10] better from 0 to infinity this doesn't matter because everything is symmetric [00:35:04.05] [00:35:05.22] if we integrate before a transform the get the value of the original function at 0. [00:35:12.05] [00:35:16.06] Transform of the forward transfer which is the original function times to buy maybe [00:35:21.02] [00:35:21.02] depending on the family's ation at 0 so. [00:35:25.18] [00:35:27.23] This center go it's not here yet. [00:35:30.10] [00:35:36.09] They send to go by 0. [00:35:42.06] [00:35:45.01] Is devalue of this function at 0 which is then to go [00:35:50.05] [00:35:51.06] over our dog and all of our course should density. [00:35:54.23] [00:36:05.12] But what is this center goal the center goal is just. [00:36:08.09] [00:36:11.15] Eleanore into the power negative when. [00:36:13.17] [00:36:16.13] It's a no and then tear into Google. [00:36:18.13] [00:36:20.06] And this rewrite in polar coordinates axes are dead [00:36:28.01] [00:36:28.01] we get this section of this fear by x. out. [00:36:33.02] [00:36:36.07] Of to go from 0 to infinity. [00:36:39.02] [00:36:40.08] Minus one and this function at that so [00:36:45.23] [00:36:45.23] one plus r. squared that the one square [00:36:49.13] [00:36:52.19] one plus that and squared r. squared. [00:36:57.23] [00:36:59.18] The r. and then. [00:37:02.11] [00:37:05.17] And this is this fair go read and transform of this function so the norm [00:37:11.16] [00:37:11.16] to negative one Norm to the power negative one is there add and transform of. [00:37:16.14] [00:37:18.00] Function Unfortunately this is not a function. [00:37:20.20] [00:37:22.14] And while I unfortunately don't know it's a fortunate thing. [00:37:28.18] [00:37:28.18] Because this isn't to go doesn't convert at some points if [00:37:32.21] [00:37:32.21] there's a 0 then it's just minus one so this is a measure of. [00:37:37.08] [00:37:38.10] You can prove that this is a measure. [00:37:40.04] [00:37:43.04] So this is their ad and [00:37:44.12] [00:37:44.12] transform of that measure which means that there's an intersection. [00:37:49.09] [00:37:50.15] And I will prove a more general. [00:37:52.20] [00:37:54.08] Fact later but this is just an idea. [00:37:57.09] [00:38:09.18] In fact what I'm going to prove later. [00:38:11.23] [00:38:14.08] Maybe. [00:38:14.20] [00:38:16.17] You want to. [00:38:17.05] [00:38:26.03] So. [00:38:26.16] [00:38:29.21] Ok if you really want. [00:38:31.10] [00:38:32.12] To. [00:38:33.00] [00:38:36.18] So what does it mean the trick of an intersection body off something. [00:38:40.10] [00:38:43.11] It means that. [00:38:44.12] [00:38:45.17] Their ideas of never direction is. [00:38:48.05] [00:38:50.19] Then my eyes one dimension of the well them of [00:38:53.10] [00:38:53.10] the section a felled by extra pin to killer but because of and wonderful for [00:38:57.23] [00:38:57.23] your formula which was written here and there with your members it. [00:39:02.18] [00:39:04.22] So it's one number and minus one. [00:39:07.13] [00:39:12.11] Plus one. [00:39:13.06] [00:39:15.13] Of c.. [00:39:16.06] [00:39:22.21] Affects. [00:39:23.16] [00:39:29.14] Because I have x. here. [00:39:30.21] [00:39:38.06] So if this is there for a transfer of that then this is the transform of the other [00:39:43.12] [00:39:43.12] one because everything is even so the former transform. [00:39:49.01] [00:39:58.20] Of this function is one over by and minus one maybe 2 prior to the power and [00:40:05.20] [00:40:05.20] depending on the normalization and the regional final question. [00:40:10.09] [00:40:13.20] What is important is that it is positive so therefore you transform its positive [00:40:20.13] [00:40:22.05] if the trans from its positive could have a distribution. [00:40:26.19] [00:40:28.18] Transform is positive which means that [00:40:31.17] [00:40:31.17] apply to any positive test function it gives a positive result. [00:40:36.18] [00:40:39.12] Because of a distribution whose full returns from is [00:40:43.09] [00:40:43.09] positive distribution which serum its attempt to measure. [00:40:47.05] [00:40:49.08] And it was dysfunctional give the result that f. is called positive definite. [00:40:53.12] [00:41:05.02] Because of an intersection by the office if this is buzzed of definite and [00:41:10.21] [00:41:10.21] therefore we transform is a continuous function but [00:41:14.01] [00:41:14.01] if we allow measures instead of continuous functions we get all intersection but [00:41:18.09] [00:41:18.09] I'm not going to do this argument so there is there. [00:41:21.16] [00:41:24.06] Origin symmetric. [00:41:25.10] [00:41:31.01] Star k.. [00:41:32.11] [00:41:38.06] Is an intersection. [00:41:39.11] [00:41:48.00] Even done only. [00:41:48.23] [00:41:51.10] The norm of k. to the bar minus one is a positive definite distribution. [00:41:55.16] [00:42:00.16] So it's very easy to check if something is an intersection. [00:42:05.08] [00:42:06.13] Not. [00:42:07.01] [00:42:09.12] You just take the norm to the bar negative one computer before it transforms [00:42:13.15] [00:42:13.15] if you can. [00:42:14.07] [00:42:17.07] If it if it is positive we have an intersection word if not [00:42:20.21] [00:42:21.21] it is not an intersection. [00:42:23.05] [00:42:25.19] So I leave this for later and there in the remaining nothing. [00:42:31.03] [00:42:34.09] 7 minutes. [00:42:35.08] [00:42:50.20] So this puts in danger my 3rd topic which is that was a month. [00:42:55.06] [00:42:56.11] Or so you're not going to cure. [00:42:59.05] [00:43:04.02] So I want to prove this in the quote. [00:43:07.08] [00:43:40.05] And we need a couple of from it was the intro go. [00:43:44.22] [00:43:45.23] In polar coordinates so we take this fear in every direction we go on with. [00:43:52.22] [00:43:55.02] Their ideas of Ok which is the norm. [00:43:59.05] [00:44:01.11] Ok to minus one Ok [00:44:05.09] [00:44:07.10] and we are done minus one half of. [00:44:10.04] [00:44:12.12] The hour. [00:44:13.06] [00:44:18.03] So this is the polar formula. [00:44:19.22] [00:44:21.03] If. [00:44:21.15] [00:44:23.17] Section. [00:44:24.13] [00:44:29.12] This is then to go lower the corresponding sphere which is now section of the spear [00:44:34.09] [00:44:34.09] by this guy per plane then to go is still up to their ideas of k. [00:44:39.16] [00:44:43.14] but r. is to the power and minus 2. [00:44:45.23] [00:44:57.10] Which happens to be this very cold I don't transform of this [00:45:02.06] [00:45:02.06] function point c. So it's this fair go read and [00:45:07.01] [00:45:07.01] transform of the function and to go from 0 to. [00:45:11.23] [00:45:15.23] The norm of Kate to minus one. [00:45:17.19] [00:45:19.09] Minus 2 of our. [00:45:21.06] [00:45:23.08] Are at the point c. [00:45:26.11] [00:45:29.07] Ok. [00:45:29.19] [00:45:40.02] Well I was holding this so that. [00:45:41.21] [00:45:54.14] So supposed. [00:45:55.13] [00:45:57.02] To cover number apps on such that the center goal is less than that. [00:46:02.17] [00:46:04.06] For. [00:46:04.18] [00:46:10.20] So this means that. [00:46:12.06] [00:46:14.09] This Fair Go read and transform of this function. [00:46:17.07] [00:46:27.23] Is less or equal to absolute. [00:46:30.18] [00:46:33.15] It's the same thing as before. [00:46:35.02] [00:46:36.14] Now let's for this k. Let's take the board to d.. [00:46:40.05] [00:46:41.14] Which is that would you give us does the outer volume ratio distance I'm supposed [00:46:46.06] [00:46:46.06] to write that it's almost gives but I will write exactly the quote just teaching [00:46:51.12] [00:46:52.13] Ok I have to write one plus Delta and then send the 0 which I don't want to do. [00:46:58.02] [00:46:59.13] So a decent intersection boy did case inside of d. [00:47:05.08] [00:47:06.18] and the volume of d. to the power is exactly this out the volume ratio distance [00:47:13.18] [00:47:16.13] times to learn so this is the body giving as the minimum. [00:47:21.18] [00:47:23.16] I'm supposed. [00:47:24.07] [00:47:25.09] To technically. [00:47:26.03] [00:47:30.13] And since this is an intersection but is there exists a measure. [00:47:34.03] [00:47:38.19] Like this and I will call it new d.. [00:47:42.19] [00:47:45.02] Ok so this measure will satisfy this equality and [00:47:50.05] [00:47:50.05] I integrate so that the corresponding measure. [00:47:53.01] [00:48:03.12] Is. [00:48:03.23] [00:48:05.12] A new dean. [00:48:06.23] [00:48:15.19] And we integrate both sides over this here because there is spec to this measure. [00:48:23.03] [00:48:33.18] Now what do we get here we get this farrago I [00:48:38.15] [00:48:38.15] don't transform of something integrated by this measure so. [00:48:43.09] [00:48:46.06] This let me get this. [00:48:47.14] [00:48:49.06] So it will get norm of d. to minus one and this function f.. [00:48:54.12] [00:48:57.17] Getting. [00:48:58.07] [00:49:00.07] Here. [00:49:00.21] [00:49:02.01] Is then to go over this fear of Normandy to minus one [00:49:07.07] [00:49:11.07] in to go from 0 to. [00:49:13.11] [00:49:19.06] X. came minus one. [00:49:20.20] [00:49:22.16] Minus do. [00:49:23.14] [00:49:24.17] X.. [00:49:25.05] [00:49:27.02] Are the x. [00:49:29.21] [00:49:29.21] So this is a function f. what is written there it's not. [00:49:34.20] [00:49:36.01] That. [00:49:36.13] [00:49:39.17] Ok so that's what we get now since d.. [00:49:43.17] [00:49:45.22] Their idea is of d.n.a. every direction is bigger than their ideas of k. So [00:49:50.03] [00:49:50.03] this is a greater equal than the same thing with k. so I will just put k. here. [00:49:57.21] [00:50:01.10] Which is not necessary. [00:50:02.14] [00:50:04.10] And. [00:50:04.22] [00:50:06.13] I could have kept. [00:50:07.14] [00:50:09.12] Him to go from 0 to. [00:50:11.09] [00:50:20.23] Minus are. [00:50:22.03] [00:50:24.09] Done minus do. [00:50:25.14] [00:50:27.18] X.. [00:50:28.08] [00:50:30.04] Are the x. So [00:50:35.19] [00:50:35.19] with this are we get very gentle things just subtracted something illegally [00:50:41.07] [00:50:42.19] so I have to add that So add it. [00:50:46.16] [00:51:08.23] And what is this. [00:51:10.02] [00:51:14.16] This is then to go over a. [00:51:17.01] [00:51:23.06] And this is a positive number. [00:51:25.02] [00:51:26.07] Because smaller than the norm so this is always positive [00:51:30.12] [00:51:32.15] so this expression is greater or equal than. [00:51:35.08] [00:51:36.10] Then to go forward and I need a couple of minutes. [00:51:40.18] [00:51:45.22] So this is. [00:51:46.22] [00:51:49.02] Greater than then to go off for work so [00:51:53.23] [00:51:53.23] now I would have to deal with this thing because. [00:51:58.15] [00:52:00.00] Above. [00:52:00.12] [00:52:04.12] And we don't need this anymore. [00:52:06.03] [00:52:11.00] So they enter girl the new d c. [00:52:15.17] [00:52:17.16] O with this fear. [00:52:18.20] [00:52:20.22] Is actually the it's a goal of one. [00:52:22.21] [00:52:24.03] Against this measure. [00:52:25.08] [00:52:27.04] So. [00:52:27.16] [00:52:30.01] I divide it by. [00:52:33.07] [00:52:38.06] The area this is surface area of the sphere where we integrate [00:52:42.13] [00:52:42.13] this is a big step spear so then why no where. [00:52:47.03] [00:52:49.00] This be except here. [00:52:50.13] [00:52:51.23] Is the red and transform of one. [00:52:53.23] [00:52:56.23] The red and transform of one is going to go over big spheres. [00:53:01.19] [00:53:06.22] So just one is one over and. [00:53:10.21] [00:53:11.22] This is. [00:53:12.12] [00:53:18.08] The red and transform of. [00:53:19.16] [00:53:29.16] The red and transform of one divided it's one [00:53:34.15] [00:53:36.19] so this one that we cut here [00:53:39.17] [00:53:39.17] can be replaced by there I didn't transform of one Larry deep. [00:53:44.08] [00:53:47.06] And now we use the same a quote days before with. [00:53:51.23] [00:53:54.20] One so. [00:53:57.07] [00:53:59.12] This equals one over as and minus 2 [00:54:02.22] [00:54:05.13] in to go over this here the norm of d. to minus one [00:54:10.13] [00:54:11.16] times for the function one dx and then we use code there isn't a quote. [00:54:18.19] [00:54:25.12] Arrays this to the bar minus. [00:54:29.13] [00:54:32.04] And to the power of our number and and the function one [00:54:37.16] [00:54:37.16] will give us the surface area of this here to the power and [00:54:42.16] [00:54:42.16] minus one over and this is called there is in the quote. [00:54:47.00] [00:54:49.07] And this is the volume of. [00:54:51.00] [00:54:54.11] Dimes and so some constant which I am not going to [00:54:58.22] [00:54:58.22] compute its a nice constant to believe me. [00:55:03.21] [00:55:05.04] Well you move d. to the barn over n. [00:55:09.09] [00:55:13.18] and the volume of did to the power by number and [00:55:17.12] [00:55:17.12] is that so d. r.. [00:55:23.17] [00:55:33.01] So the right side. [00:55:34.22] [00:55:37.06] Here is smaller than absolute times what the just good. [00:55:42.06] [00:55:46.04] Sees less than one technical. [00:55:48.13] [00:55:51.01] And over and minus one. [00:55:52.16] [00:55:55.15] Ball. [00:55:56.04] [00:55:57.16] One over n. o. the lead in bull and dimension and minus one [00:56:02.11] [00:56:03.20] and this is less than one and this is less than 2 to make a rough estimate. [00:56:10.10] [00:56:11.15] The constant is less than 2. [00:56:13.06] [00:56:19.14] And then. [00:56:20.04] [00:56:21.05] Our. [00:56:21.17] [00:56:30.17] So as long as all sections. [00:56:33.18] [00:56:35.04] Absalom. [00:56:35.18] [00:56:36.18] They enter go off half over the smaller then absolute [00:56:42.08] [00:56:42.08] times this so now all they have to do is take up so an equal. [00:56:47.03] [00:56:48.11] To the maximal section. [00:56:49.21] [00:56:56.14] This satisfies this inequality for every direction obviously so because of maximum. [00:57:03.13] [00:57:05.00] Distance volume and we don't care anymore and we get precisely this in the cool. [00:57:11.13] [00:57:14.09] Ok. [00:57:14.21] [00:57:16.04] Thank you. [00:57:16.17] [00:57:19.12] Thank. [00:57:20.05] [00:57:29.01] You. [00:57:29.13] [00:57:36.05] I don't know why do. [00:57:37.06] [00:57:39.17] You get this. [00:57:40.07] [00:57:44.16] All. [00:57:45.04] [00:57:46.12] Convex there is joins there. [00:57:48.10] [00:57:49.23] Ellipsoides are intersection but it is [00:57:52.00] [00:57:54.08] why because the transform of the cli didn't norm is the norm and [00:57:59.08] [00:57:59.08] if you take a linear transformation it's the determinant So it's still positive Ok. [00:58:05.09] [00:58:07.12] So it's by joins. [00:58:08.13] [00:58:10.21] This is smaller than the square root. [00:58:12.10] [00:58:14.15] And the square with the fan that there is to be able to move but for very special. [00:58:18.14] [00:58:24.08] So I'm going to consider different classes of k. next time I [00:58:29.08] [00:58:29.08] will start the I will mention everything that they know will not do anything. [00:58:34.08]