[00:00:06]
>> Ok so. Thank you that's the talk let me. Remind you some of the things that I was like in previous time again I would I would say few things at the time you do mention previous. And let me introduce Libby deflation that I'm going to use today so let me explain what I what is written here let's say it's met a complex body so.

[00:00:36]
The support function Ok and we can prove true events into the consideration of measuring the Goshen space so that this is not the Goetia and then the support function concentrates around this mean it's gone just the diameter in that case so the concentration of mass of say that I have on that level and the case thought of see this exactly this rescue this immediately follows from the spacing of measure in this rescue is the voice can number all of the dual.

[00:01:11]
Ok we also mention. This follows from other result of the Converse seen in this is now a small ball probability for the support function. Probably the support functions there's an absolute times expectation behaves at least like absolute decay stuff Ok. And. What immense Imprimis not only things that usually this is not the right estimate you have something better for the small boat in general the support function is a complex function so you expect that the small ball will behave better than the.

[00:01:50]
Concentration so I'm going to use these things not on the on the Goshen space but on the sphere so they expect a certain on the sphere if they so this is just one of these I'm in with so the cumin with so I think rate of of the sphere the support function of the q. and I was going to use it for positive and negative values and.

[00:02:17]
Doesn't do Grayson bipod to connect they need to go out of the sphere within the guts of the Goshen So this is the connection so if you just use these 2 theorems then you get the following that for all points for all cues do you. What's going on but that I'm mentioning that moments remain constant constant up to the expectation which is the main with Ok So this follows from.

[00:02:44]
From that point however what is written here it is very balanced up or these. Moments in the negative moments behave the same However we know already have Desmond's in it again that the negative moments behave better than the positive moments and we have also a better level of concentration.

[00:03:06]
So I will come back to that this would be my my theme today so let me remind you that for. 4 complex functions. We prove this inequality and this leads to the following corollary. I think that one. The probability that half of our normal office support functions. To be less than epsilon times that expectation.

[00:03:50]
Behaves like you want to. See over. The sea were we don't. See is the variance of the support function or. Expectation and this pond it did this month smaller This is what I explained previously Ok. Ok so I I I want to say 2 more things about. This theorem 1st of all also to connect the things with that.

[00:04:38]
Talk of Siri you have if you willing to you move from the violence to a big quantity then the statement cult's true not only for the Goshen. Space but more general So let me write down the statement let's say that they have a measurement you. Satisfied is. What they sent.

[00:05:09]
That's what they sing inequality. Which means. In this sense. Rather the ventral be. So. I prefer not to say the day this but if it do here it is one on Syria talks you take c. to be the function so this corresponds to. The w. 2 and this is an inequality that already Siri mentioned of course many measures of this one particular.

[00:05:59]
Goshen and then you have that you have a similar statement and f. is again. You have proven it the effects of that lesson and minus c. Time Square Root now you would be right in. The expectation of the median number of the Grad squared this is less than one half of my news spread over 2 so if you if you prefer to you if you want to you buy this price said of the violence this is a big you want to buy punk or an equality that form my zones that are much more general than Goshen and you have this inequality.

[00:06:44]
And I want to mention also one more. Theorem. So of another class of of that once a to consider is the class of. So we have already mentioned the concave but sometimes so I mean the one main and on the time same as number times b. is bigger equal than the ones number.

[00:07:13]
On them so if I have a look. Then you just mimic the proof that. I meant the Goshen case and you get the following so. You get the probability that f. off for. Fall. Say this he did with respect to you that's a probe into Mr no less than and minus 3 times the Screwed of the variance then you have mines the.

[00:07:49]
Squad but that means. The proof is completely similar if you want you just remember the proof of this one you will practice for this one and you will convince yourself that 6 actually say. So you have these follow concave meses that. You have a similar inequality and you have indeed.

[00:08:13]
The variance and it done so you cannot hope to have this squared because here is the example that I think it meant and also very quickly at the very end of my talk. Take. The measure them you. Do have then see the indicator of. The end for me is the ball of volume one.

[00:08:39]
And they have to be that you can you know so if you will take this abuse in it's just equality because you just compute and you have that bound here with this it reminds the so that means. That means that even for measures that they said if I look you cannot hope to have this done I hope to have this is the mate and you are back to the gate so.

[00:09:12]
This is this is to be. So now I'm going to move to my 3rd part of the talk so I'm going to discuss exactly. Case. So now you will be a locum cave. And I would like to understand concentration on. Measures So the goal. As a so you will be always looking.

[00:09:54]
Of course there are many questions that have already. Mentioned So I would come I would try to connect also what I'm talking about with. The other dogs so I will try to understand the look of measure by understanding. The tree of the month so in order to you to do that we introduce this is that of of these so now be.

[00:10:28]
Given one eye and to find. The support function of the body is by definition. Expectation of them in a functional. View. To one of the beat. I mean Iran Ok. So the support function to the direction this exactly does the big moment of all fall the marginal So what we can say about this both this we have 1st of all that if used as a topic.

[00:11:10]
That to this completely equivalent to say that this it do for you it is a b. to end right and we have that they increase because of the. If you want the one that follows from no concavity implies that they cannot grow very fast so if q.. Is more than p. then visit q. of communities.

[00:11:43]
Like that. And of course I have held the so. They can look. Inside. And already mention. Already some docs about because of so. One very important example of Moses is when the new Is indicator of. So in Dr case. We also have. The k. was a one I would write Ok i would mean that this indicator of a conflict sporty so if p. is more than an then this is already has a morphic to Kate so this is really nice application of group principle.

[00:12:43]
So what is the big picture is if. I have a k. this is. Inside from 2 of k.. If I have just indicated of k. then basically in that case that. It is a multiple of the ball and this is something. So that I have an increasing sequence of bodies so.

[00:13:15]
This. When I arrive to use it done. Then I have already come up to. I have already arrived close to my body k. Ok so to connect things a little bit with. Elizabeth duck Lizabeth has already introduced a family of comfortable is the floating boat this there are again a sequence of bodies that they're small and they create a cape and you can check that the floating boat this is.

[00:13:53]
Isomorphic to you something that people is when I think could be logged one of the public as Long Beach is less than 0 the picture here is. The floating bodies our eyes are more fit to this world is this connection of moments and days. But what Elizabeth was talking about this we see was mainly interest when I'm approaching that boundary and one imprint in the boundary and then this thing that bends on the boundary but when you are in time the convertable do with this and then this depends on the distribution of the volume Ok so these things are very very close when you are you test the you know what you are inside when the say is not is not 0 and you will expect to be different when you.

[00:14:47]
When you arrive to the boundary so this is a connection. Ok so what they can say about about this ball is what more I would need. So also. Here I'm bussing I draw the picture for conflicts but this it is sometimes useful to want me to measure sometimes it's useful to work with complex bodies when it is useful to work with masses you have immediately the following nice program that if I will take the prediction of the b. of a measure then.

[00:15:32]
Does it be over the months of the measured. Ok so this is this is very useful So this is that when they put expose this I just see this it is it's a see that be of the margin of. Now because of these it is also useful to walk with with conflicts bodies so this is the point that this ball this of goofball that Syria already mentioned.

[00:16:00]
Are useful so let me. Introduce it right now that. Of. The measure it is. In Iran. So that cubing to grow from suit an infinity. Of fall our own feet. To be mines one. Bigger equal than half of 0 Now if the density of the measure. So I can introduce this this bodice when you have look and give it to you know again that this ball is they can not be very very different for being Q So you have.

[00:16:46]
Let's say that use also symmetric. If you look on cave you know this body is convex So that was the theorem of both the film of Guzman and soon has already mentioned it and says that the recent months and of being q.. When you have the Liz and.

[00:17:13]
It's not important to you to write don't do you right now what consent but this is really about one dimension you won't be about moments of looking cave meses Ok so you just check that there was a case and I read this when there are. Exponents Ok so and then you compute the moments Ok so now that you have this you can sack.

[00:17:40]
When you have this inequality you can text does it be full. So when I'm writing this in summer while working with dimensions. Of mention it my my my my agreement is the case volume one because I agree that they have the help of any right so I'm working with pro building measures.

[00:18:04]
So this is that the be of the. It is up to universal constants the f. with 0 zip. Ok so. So you can always from. The measure to you. That would be the key. And vice versa and this is. Up to you to both of us. Ok so immediately because also of this we have.

[00:18:49]
K. And last few of you. Like that so when I wrote and plus I mean the list you divide by its own. No I know my lies do not have one so that's the so when I have this is it then then immediately I have this it says I'm off to make a entries through.

[00:19:18]
This is the principle. So if only. To wander around I have a this is one and this gives the following fact that it is very useful. All Fall of mines or it is one or. The volume. The value of. The value of the dent at the point 0 So I'm working on the smell case and.

[00:19:58]
If you sense of topic. That means. The definition of as a topic on something you it is exactly that. Ok so. For every is a tropical wave measure this is that measure all fall but it is one measure of this event both this one over tropical and. Ok you know what else I need.

[00:20:35]
Let's talk about the following can sequence so let me apply this to you. So this is true for all of those something that has it has mentioned already it is that the margin notes of local cave measures are also located I learned to inequality so if I would just apply that I get that Zed.

[00:20:59]
Is it be. The projection of the zit. You. Do one of the key it is one over. The valley do one over this is through 4 looking cave so if I will apply this. For again then Decatur of keys of it may go back to this again then I get the following.

[00:21:34]
The volume of. Here I want the ski. Ok I will apply this for people skate so then and if. So what I'm writing here it is that now this is k. dimensional since I am on the projection so this is equal to the k. a for k. dimension on.

[00:22:10]
Provability So by this remark a sequel to this Ok now I will apply to 4. And they cater of k. So now I have the projection of that. Small capital but this is not confusing this is one over the monsoonal of the point 0 but the bugs and when they have a conflict spotted this one.

[00:22:37]
Intersect. But to one of a k. if you write this once again this is what I was calling the roots of support to connect it. With talks about the support this is that the k. key to one of the times to say. To one is. Between 2 universal constants connected with the roads us about think about the following Now if I will instead of This Is It.

[00:23:17]
Then this is less than 2 skin and less than one so I feel you will replace how I will make this equality to be solved you replace the k. with exactly this it and if you see the picture it says that if you are in key demands on you have to go to the level of that and then the quality becomes sop up to Universal.

[00:23:44]
So this is the. The separate inequality and then. I will promise that they want to talk about cosa trace and that they want to use this bogus in order to understand concentration so let me start with the following let's say that I start by integrating. So I go back to you measure those you love.

[00:24:20]
This 1st computer has nothing to do with love. And use that Ok. So let's say that I integrate on the money and z. and k. for me this is the most money and I integrate the. Constants or if you want the margin of the point 0. Great then with respect to the harmony.

[00:24:51]
Ok let me make a computer so this. I can write it. I can write it into Gollum's he and Katie and let's recall what is the. Point 0 that means that. If. Integrate my functions that. I want to. And then I will write on all unfolding it so I have some goals on the defense and Katie and I have.

[00:25:31]
To go all over the grounds money on the integral over the sphere. On the half. And I have out to the Kamens one half of our. Time. And a half. And I have to have. Ok. Integrating over the girl's body and it will be integrating over the Guzman in with the skate so I can and have also made it with stuff you don't Ok this isn't.

[00:26:16]
So if I will integrate over the breast bonding and then on on great spheres it's like integrating a big sphere so this is I have an integral over the sphere in the grove. From 0 to infinity in the have Indigo over the sphere have some points and that I would compute at the very end and I have to the Cayman spawn follow.

[00:26:42]
The r. I Now if you write again what it means and we do it again the polar coordinates you get that you have some constants and I have the integral in a run and I have one over. I. Don't know if you were. That was. Here you know you're right you're right.

[00:27:15]
So. That it would be Ok I started with this Ok so let me right here the formula formula sais that if I will take that expectation of the you know to the mindscape in the minus one of this is up to a constant Ok. Integrate the integral over the grams money and all the.

[00:27:49]
Ok this is what. I said Now maybe this looks. To those not to the working on the dual. This is nothing else than saying that. On the sections it is the case do what makes it so that it was a competition so you would see that something similar Ok so but now let me start again now that was nothing to do with it it was just compute a sense so now if I'm going to use a look.

[00:28:29]
I'm going to do the following. So the minus one over case this is equal to the integral of the grasp on and one over the projections of k.. Ok I just use was it. Ok that is true point wise. And now I'm going to use some dialogue and I'm going to use also the reverse and the low to write that this is equivalent to to go over the gross mine and I will have to answer the computer.

[00:29:16]
So this is the section of the new. And if I would write no this one going to needs then I went into the responding and going over the sphere. And here I have one moment the support function of that. Of all you point. This right this one and it's Ok.

[00:30:00]
Mine's one. But then again I would do this together because this is a and I have this is the integral of of the sphere the support function of. Of the power of the mindscape minus one of a k.. And I have some constants. I can of course keep track of them.

[00:30:27]
Once so. No use also look. I proved that this is. The w. minus k. k. of you. Have to keep track of the Kong sense and the constant subsequent offend of Ok. Ok that does fall into a bowl that you compute so the w. minus Kate is the one that introduced at the beginning if a degrade over the sphere support function to some power will write it.

[00:31:03]
Ok so this is the 1st formula. And actually you have the same form rules for positive. But this is trivial. So. If I would start with. The w.-k. of that gave you and write down what it is this is the goal of the sphere integral of all phone in a functional.

[00:31:35]
D.. This ng I think. To want to keep right is what it is so going to do for me and that's compute what is here has nothing to do with more so this is the final k. but I will have that you do know so this will be an integral r.n. the Euclidean on.

[00:32:04]
The next one with. You. To come this is. Sent and a little bit. And for me Ok you can rough those you can compute all the constants about because I'm working also for negative moments. I don't have in the ability for so let me write it also here.

[00:32:32]
They said nothing you need to know k. one of. The final k w k of this is that you Ok so the point what we've done is the t.v. is that. Here we have some to go with the spec to our local. Capability. But they pleaded and we managed to do have it does.

[00:33:01]
No matter which is the sphere but of some strange function that the Ok so. Own time was. Ok So Ok so now I can write down the theorem. If you. Don't pick. And welcome cave. And. The lesson school to find. The negative moments. Up to universal constant always positive moments.

[00:33:57]
Ok let's see wife. Well 1st of all so let me run the proof you. I didn't want to face. Ok so. Things will be equal to that. If and only if these 2 guys say. Yeah but I haven't been here so I can play. I can taser guns and hit with a price this is what.

[00:34:54]
You would see from the proof. Ok so it's a. Constant holds. No no no this is a universal promise and yes but of course the most powerful on it is the law. Ok now I will try to explain. Again but for the moment I just need to say.

[00:35:21]
Yes this is an extra minute rounds this should be like that right the moment is related to my so Ok so now we're ready to prove it. So. That if you. Do things will be equal for all be. For old b.. W. mind speed is it if you can you will be the w.p. you.

[00:35:56]
By the consecration in this this results if. The will be less than the. Number of the zit you write if this would be true then then I'm done so I need to understand how the device can number of Zip be behaves so. I raise it. Let's say that this is b. and this is the number of the.

[00:36:28]
So when do than we have. A ball so I'm starting piece of ball then I have the numbers said so then we will start to drop and this inequality will be the 1st point. These things will be so that words can number. Of this it be it is expectation of fall off is it.

[00:37:03]
Over the diameter of the. Square Ok this is this is what I get from concentration great but I mention that was the wrong white race but I mention that. Is that don't increase very fast and can increase its linear spec so this guy cannot be more than p. times.

[00:37:30]
So. This will be less than in the here and I have Square and now I was along. On this level then this will be equal if you also the moment so I can replace this with a beat of commute to the be and then to over. That's right action and now I can use.

[00:38:00]
I can use again this inequality. And relate this to you. So I have a spurt of fun because I will I will move from Goshen you. To the sphere and then I will replace the square inch. Of. Her round. And this is squared and here I will have the expectation of that you can you know.

[00:38:36]
One of the squared. This will be. Will disappear I will just do a health. And I will have a b. from the Spoke to have a piece here and I replace things with the norm. To the square and because of this is so for the support of this is what.

[00:39:04]
So I have and so this became said at least like and. So I want the best point that b. will intersect and over p. so it will be fun. Ok so this is proved right that's that's a proof and corollary if I would just think of cause this is.

[00:39:31]
Where the fun is expect a certain So I have. To know when to be more than constant times the square of the fan. Will behave like a minus the square of the fan this is a simplification of Berkeley. And. The Gnome is less than 10 square. Is less than.

[00:39:58]
The square of. Ok so this is. The 2 results. So. I think this is a complete proof up to some calculations that they ski for one dimensional functions Ok so. This is for a bit. Bigger than long and this is for him as I. Have it. Now to remarks and then what is going on this inequality fence out to be sop it's going to be improved That's a case of exponent so this is that.

[00:40:46]
That we've done a lot of the vase and resume. Is The Right One In this approach what. We this is about this one probability again we've done mistake in some sense you prove a small ball by connecting. This is like saying that small ball probability it's at least as good as.

[00:41:10]
Happens all the time but I say explain previous time for complex functions you expect something much much better so this is not known to be true. Ok it's not done to be true and. This is let me explain how this would connect better. With. The thought process so let me again.

[00:41:41]
This is the. Things we. Have plain conjecture so the conjecture is. That the. Exact same thing below is an absolute Time Square the plans less than c. times. But. Ok. This is completely reliant to the hyperlink and. Now it is we don't know so for a false war balls but as I say explained on the copy in the small boats are related to the duel theory with very much related to the section bodies.

[00:42:32]
So there is a close relation and. I'm sorry x.. X. is isotropic. So this right this this result if you can make the front and you saw it and not only that if you can make. A new one half plus epsilon then you prove the 4 to 5 so the fan here this exactly because the best bound for the fun so this isn't don't work with and it was not nice.

[00:43:20]
And they connect an end and. I have to say that they quit Millan's it is that if this is true for all. Then they have to follow the other direction of course is almost trivial. And there is one to devotion to this one with says that if you can if you can prove here anything better than.

[00:43:53]
The full and this is the mate gives the fall through the pull of. Ok and. What is wrong Chris what is wrong with this proof the wrong with this proof it is that it doesn't take into account that. The proof goes through also the moment it's approved base and concentration and not directly unspoiled will probably.

[00:44:26]
Ok how much time at. Ok. Thank. You. So. When q. is more than that on the bows of the side you know exactly what how it behaves This is a result of 4 sides for complex result of. Personality. And maybe one second. So you know exactly on the pose the woman's how behaves I didn't wrote it down because that was I didn't need it for my proof it does work down but this from a proof for the negative moments no nothing.

[00:45:35]
Negative almost is a small ball problem and all the problem is that we know nothing. In the Goshen case. So the result that they mention about the result with that was well it doesn't cause at the committee if it is the 1st quantity of result about the small boat probabilities that I know of in the Goshen case and for norms.

[00:46:01]
And. You can take another and you can so that for every regular that you may hope for there will be an ellipse so that fails for small bore. So that you don't have any good regularity like you have on the on the positive ones. Like condition. So if you know for a fix one you know that they have a plate with 6 or is 2 then you get this for free.

[00:46:44]
Because you know that the density is bounded So this follows just by my inclusion not all the problem is the so that if if you knew how to do it in a point wise month or the new again you solve the problem you have all the usual things so you need to have it for for all of you need practically this goes through I'm positioned so you need to know that there is both a that.

[00:47:11]
Has a topic it's also in position and you can construct I was possible both that you can put it in both positions and then it follows because the negative moments and level and. This is. Very much related. If you think you. Know I think. Thank you very much.

[00:47:48]
I can construct some very artificial but thinking about this and brought them with something big that they know the hyperplane conjecture but not enough to know that the side slides but. Only in a different cell become. Part. Of the small ball is the problem is that playing jokes this but yes that's a problem as you say if not for them and I think you can go to your home.

[00:48:22]
So that. You have the. Given number. Ok I don't like it giving up on. This let's call it. Ok so I think you get this. Some. No. Ok Ok Ok so let's Thank you Daryn.