[00:00:05]
>> So what I wanted to show you it's already running it's on its own good so what I want you to show you some pictures of the floating body so here everything so that's. The cue and everything. Goes well here we have no problems with existence it's 0 symmetric so we know the floating body and the convex rotting body coincide so there are no surprises but then if maybe we look at some other pictures as I told you.

[00:00:45]
This one. When we have. So now here we have an octagon and then things may go wrong at some level so here you will see these kind of weird shapes appearing if we do the construction by looking at the midpoint and. And so the floating body may not always succeed but the convex floating body always exists.

[00:01:14]
And even if the body has some small knows that there may be potentially a problem as you see. Part of the not so how far up or so facts are these pictures I wanted to show you and the credit for these pictures goes through all the n.b.c. it's nothing to do with me so thank you.

[00:01:41]
If these. You know give us. A miracle. Ok so now let's continue what we wanted to show was the theorem. So I write a tear. Each said that. If we look at the volume difference often Abhi Tracon exploit the Cape and its floating body. And. So to speak take the derivative to properly adjusted derivative of the volume.

[00:02:30]
Look at these lean meat goes to 0 then exists and is equal to the into going over the foundry of cave where we are heading to the One Ring plus one. Point x. and we integrate with respect to the usual So if you measure of u. k. and this is the generalized curvature that we have here we had done.

[00:02:55]
We had arrived at rewriting the expression. On the left. One knows and I have explained that it was the limit. Goes to 0 of the integral over the boundary of p. and then we'll have boundary point x. in a productive it's all to normal and then. Came in this quantity of all there is just pure but none came in the quantity.

[00:03:30]
Over x. so that the respective u.p.d. and norms raised to the power in d.m. u.k. x. n. peace if you need it you need to be reminded of what everybody of course is so far. So then. The goal was to move inside and you know order to do they want you to use a big dominated convergence theorem so what we wanted to do is we wanted to see if this function uniformly and does have a quick point by c makes buy and own form of off by an integral bow function and the can get it there was a rolling function not blush to only function.

[00:04:14]
So to speak what we just call rolling function which gives us an analogue of. Off the Block a rolling rolling ball so the rolling function which I had called out of x. was simply defined as the mix because actually there's 2 but this would be said taint of all row bigger than 0 so they the ball.

[00:04:40]
At. X. is a boundary point but then I look at the ball that center x.. Row and next. And has radius row so that this ball is contained in k. and they're supposed to happen if an x. is unique. And it's 0 otherwise it makes it so they push these rolling function and it's into in terms of this rolling function that we are going to try to bind to bone from above these.

[00:05:22]
This expression here under the Intercoastal I'm breaking these things to your room. So that those who've been to past. And myself and read what I said these quantitative worship of the last grueling fear him so what does he say so we'll take a general can exploit any real need like scaling the right scaling so key such.

[00:05:52]
You need or is contained in that's just for appropriate scaling which you already see in the next statement that you mean One such a thing so then for all. Between 0 and one we all have of various things namely Firstly they said x x in the boundary of piece so it out of x. is bigger really well then t. this set is closed.

[00:06:25]
And that is important because from then follows that these rolling function is measurable which we need if we want to do something something to group into believe the properties of these warning functions so this implies they are is measurable. And then comes the next thing which is the most important thing so the measure of all those points under boundary.

[00:06:57]
Are of x. is people reporting t. the measure of that say is large is bigger equal than one minus t. to the in minus strong and I'm just writing so if you say area of. So from there it follows in particular from this inequality because tech so to speak the distribution function of these rolling functions from that it follows that in particular.

[00:07:30]
The function are. Let me just to make clear that we look at it as function x. rays through the poem minus Alpha is Integra bow. Before all fall between 0 and strictly smaller than one and also the inequality is optimal. To me. So let me 1st comment on this last point Dana qualities of to moan so I'm I'm.

[00:08:16]
Indicating why that is the case so look at the Pew Center eat it 0 and you taste cycling to and then so there is the infinity Center site linked to and the so now we want to see what surface area that works so can so that we can put all of radius t. in sight and we see Ok it works if this thing looks like so.

[00:08:51]
If we. Put all those points on the boundary or fall for the Cure with side lines to minus 2 t. it will be Ok. Yeah yeah. Because we go away t. here in t. here and originally we had cycling's to still be great. To work on the sidelines.

[00:09:18]
To minus 2 t. So we will see that the set of all x. for which our of x. is bigger equal than t. has measure but they say it actually equal to the. Measure of the q. with side length. 2 minus 2 teeth and that is one minus to the end minus one.

[00:09:52]
Area of the to my side lens 2 so we have equality for. The case of the cube so this can be improved in general and also the cube is also an example to. Use almost any example. That we can attend. I thought it was one in general so this is strictly in general this is strictly smaller than one.

[00:10:25]
Does not. In general. Also I want to show their true in this inequality there isa Yeah. Sure. But. I never a computed for I never computed you for the simplest. So from this inequality Darry so nice. A nice thing that one can deduce quickly and I wanted to show you these.

[00:11:07]
So what we look at so let's look at. Mukasey of the sect of Aix in the boundary of $430.00 k. So they are all fakes is strictly bigger than 0 so this we can write this u.k. of the states of on x. in the boundary of p. such that our of x. is bigger than one.

[00:11:36]
And we will have to take the union p. goes from one to infinity. And if to make you Ciancia I call these e.p.a. then what we'll get by continuing to measure that this is the limit. And goes to infinity of view of page. And there is Spidey in it for you t.j..

[00:12:05]
Well then they limit this goes to infinity of one minus one over. To the in minus one. And then let me just try I say it over there so if you say area of the boundary of p. and that is of course so if you stereo the boundary of k. So we will see the measure of all.

[00:12:31]
The surface area measure all those points on the boundary for we are of x. is bigger then 0 is actually the whole. So if you say the who sank and now northeast if we have a point already seen in the definition here and if we have a boundary point for reach our of x..

[00:12:56]
You do you then we know in such a boundary point the outer normally is unique so we'll get from these immediately. Thanked the measure of all those points on the boundary of p. for which we will have date. Well let's say like so and of x. the l. to normal that's not to.

[00:13:31]
The measure of 0 which you know. Shows Fos prove they are the boundary of a convict body is almost everywhere differentiable. That I wanted to say to. This theory and now behave which brings us back to. What we want to prove here namely the limo saying state. Positive constant sat there for x. in the boundary of pay with x..

[00:14:10]
And for all Delta we'll have this expression that we have under the into. This in a product to normal divided by in Delta to the tool room plus $11.00 minus. You know just being all for the heights of these coupons so that this expression is smaller equal thing this absolute constancy Then comes the 0 rolling function are all thanks to the power in minus one over in plus one and so this exponent for the rolling function is strictly smaller So this is strictly smaller than one this is integral.

[00:14:56]
And we have founded our exploration under the intricate technical function so we can apply dominated convergence theory him and into a change integration and limit over there. So be can actually write like you raise here. So we can actually write we can pull the limit in. Can write that this is the intro and then the limit as delta goes to 0 of this expression.

[00:15:34]
You know. So now what we have to do is to work through their on their expression here and that is another name I'm not going to write down I'm just going to tell you that working harder on this expression. Is looking at but the slim it gives us it actually does give us c.n.n. this constant That's always the same.

[00:16:00]
And then comes the generalized. To holler and plus one so that proves the theory. Now I want you to say one thing about this limo. Even if an indication of the proof or. So it's still. Not even escaped just an idea of proof of Lima star. So how does he go.

[00:16:42]
So what's the situation situation is as follows Sylvia have a boundary point x. and beyond have remembered that was how it was done so somewhere here is 0 and how this external time was done it was simply at the intersection. Of the boundary of the floating body with the line segment.

[00:17:07]
They're dealing Ciro to that's how we got the extent. X. and x. total linear so they tells us that x. can actually be written as a tell time so x.. Can be written. O. Euclidean norm plus x. minus x. don't Euclidean norm so what we are interested in these quotient so we'll divide by x..

[00:17:33]
So we'll get one is Extel turn over x.. Respective Euclidian on plus x. minus external time divided by x.. So but not we're not just interested in that we want to raise it to the power in so we'll get. What that. Over x. rays to the Polar in is equal to one minus x. minus x..

[00:18:06]
Over x. and that guy raced through the power and then we want one minus to make sure that I'm not these things. Because you messed up so I switched the inequalities in the wrong direction so far we are good but we haven't even used an inequality but so.

[00:18:29]
You know the thing here is bigger Well then one minus n. times this term so this will be because we have minus signs small or equal then and the one goes away so we'll have in. X. minus x. style tongue over x.. So that's what we would be getting And no I combine it be the other term so what we are looking at over there is actually what our expression is so let me write let me put it here and it's base here sorry about that and let's put this like so.

[00:19:12]
And below simply want Ok let's write this 1st so that will be smaller equal thing. Which shows us what So Jim Mitchell This is the all too normal shows us that this expression can be ball and you got bought by Ok they'd lend you so. That expression here is exactly this lens so now we'll have to distinguish cases to prove the 1st case would be that Extel toe or does a touch so small.

[00:19:51]
That this lens here this lens here is actually smaller than say one half. The. Radius of the rolling ball at that point so there is a 1st case and then what we'll do is so real put in our own or. Still be a pretty hideous blue. So we put it all Roni opening function on all that's coming to us from the rolling function Sylvia put that in and then we will still have to do by Delta us over estimate the delta so we know that there is a hyper plane going there is a hyper plane h. going through this point on the boundary that cuts off Delta and we know what kind of of the body is Delta.

[00:20:48]
And by but we'll do instead instead of working in the body where we have no real control of what's going on and where we cannot do really computations we work with in this rolling ball and then can compute and get the estimate from above in that case of course yeah of course Yeah but you know.

[00:21:10]
So so that's the 1st case and now there is another case when they don't reach the speech and they know he's too good for but there's another case that we have to do because basically we have these here for all Delta So there is another case we have to do it when Delta is not so the goose expression falls within the rolling ball that falls nicely within the rolling ball and that we have to handle.

[00:21:39]
So that was to give you an idea of how such things are proved. Now. So now basically I have said. Most of the. I have given you most of the ideas of the proof of this theory so let's look at some other aspects connected with it and the 1st aspect connected with it is that I wanted to explore a little bit further the situation of what's going on when we are looking at Poly told.

[00:22:18]
Because when you're looking at Paul you talked then this right hand side because. Most everywhere so the right hand side is 0 and consequently for party tops the left hand side goes to be 0 so I want to illustrate the point again by looking at the. Ball but now just in Dimension cool because I'm so just in Dimension 2 and I said that we can compute actually there how the boundary of the protein polytope looks like it is one minus Delta over 21 minus x. so we can expediently and in dimension to compute w. difference so we don't get these feeling here behaves like one can actually compute what comes is what.

[00:23:15]
Is to a Delta one plus. 2. Plus 2 Delta one overdose. Ok so then if we do give right. It. Volume difference and we divide by what re are supposed to do according to the theory him then mention to you could be due to the 2 or 3 and that still 10 goes to 0 or if we see here close to 0 as we had expected but maybe that's not the right thing maybe we shouldn't be dividing by that one because the it's not giving us the right order so if we look at the expression maybe rather than dividing by that thing here.

[00:24:16]
We should look at the volume difference and divide by Don't tell. One over downtown and then and Stelter goes to 0 this actually goes to 2 so we'll get something different from 0 if we divide by 0 so to speak the higher store on the right order of the expansion of volume difference and there is to their taste been ripped in general for every point he thought by cost so it can't just be a clone.

[00:25:12]
So what real cost and showed us that piece of all it took then the limit. On the volume difference of key and its floating point Yuto and then comes these you know why don't we don't invent to the par in minus one and always do you know this exists and it's equal to.

[00:25:39]
So this exists and is equal to the number of flakes. He and sometimes also I'll explain in a 2nd what this is of play off like what Also I'm sometimes tolerable so equal to the number of powers of p. and then in to the n. minus one into Tory.

[00:26:04]
Yeah. So what does the flag. The same some times it's also. All right tower of p. So you related to the community structure of a polytope isn't into a full. And 0 and one and a minus one where. I is and I do mention of. Face of t. and we have an inclusion relation.

[00:26:43]
And plus one fall I so. So on the how in the case of the party tops the boundary structure comes in so one might be tempted to take the number of flakes the number of towers and the right angle off I find so if your area because that must be and so if.

[00:27:09]
So 1 May be tempted to. Declare these to be the if and so if you say of a pointy top but there are other approaches one that we are exploring now we've caught and it's not clear it's not clear if they coincide What is the one that we should be thinking.

[00:27:34]
That that's the continuity is always a problem yeah exactly yes. And of course one would like something like that they close in on another approach that we do with one through their use various ideas floating around and the thing that we were looking at is actually a continuous so to speak.

[00:27:58]
And so if you say area which seems to have also all the other nice properties that if I answer you say. If you look at it you know what we have here namely that for one. You know should be dividing by the power of Delta in 4 other we should be dividing by that power of Delta they lead to a question namely.

[00:28:27]
So that and they do so also not always for Delta small so we obviously are of date the volume different behaves like so dull touch more going to 0 behaves like how so it behaves like a Delta 2 the 2 will ring plus one if peace moves. And it behaves like.

[00:28:50]
One over a long one over Delta 2 the in minus one if case of polytope. And those functions so the question East. Just for every. Comp a function. And. Function and then maybe one maybe you want plus additional properties. So for every compare function f. such that f. sits between the 2.

[00:29:29]
So such that. Ellen won over time. To the end minus one is smaller equal than a full Don't smaller equal and down to the 2 over in place one does there exist where I'm at with my sentence. Does For if you. Explain the. Key they would be depending on if.

[00:30:07]
Such they key. Difference with k. Delta East or don't I'm going to 0 like the functioning. And so we know one answer so we know an answer only in dimension to. Have we don't know. That 1st cause. So that's another question and then I want you to believe.

[00:30:45]
It's true in Dimension 2 plus additional properties. So that we can lead to some kind of partial differential equation and one can and. So. So yeah maybe I'll go there. For Jewish so. So I want you to finish these. On floating bodies by giving some for the. Outlook what has been done and in which direction what can go aside from you know this question that is open so those further subject of discussion.

[00:31:36]
They are as follows so one for one very smart only the floating body there is also something like think we call. It I call it the way to. Which I introduced a long time ago and when I introduced the idea of course so the floating bodies quit for extending the if and surface area to all convex bodies.

[00:32:08]
And very soon other notion which I'm not going to talk about which is the p.f. and surface area and so I introduced the. Floating body because it gave us a way to extend the notion of p. if and so if you were to give a geometric meaning of.

[00:32:27]
Surface area of expertise so why do you use it's basically the same idea as the floating bodies still via have a clinic sporty Pavia look at the high school being apes and gives us to how species 8 minus and each plus and. For the. Way to do so we also have a function f..

[00:32:49]
That is positive and continuous maybe and yeah and then this way to reach depends on the function f.. And before. Introspect over all how species h. plus but now you're not cut of Dotel I'm done talking to beat. So what do I mean you look at the interval f d x h.

[00:33:20]
Minus into k. to integrate together with this weight function. And so that's one. Thing that one can look at and as I said I introduced it to get to this surface area. Gym metrically I've been lately the object body has turned out to be extremely useful in extending the concept of floating body and one contrary say f. And so if you say area because there is no a fine invariant in that context so that why we call you true occasion areas so slow to extend.

[00:34:00]
The concept of floating body. And a fine so if you sorry I'm putting an find in parentheses because there is no as I said invariance. To spirit and how in type of poly space. And that was done with Florian be so. That the 1st thing and then the other thing is another concept nation.

[00:34:42]
That is kind of dual to the floating bodies so what we'll do with the floating point do you use real chop off tract can go to what one does with the illumination or do you want us to do all operation x. on heads of Wm Delta In other words to give the Think definition the illumination body which I call likes Ok delta on top that's the set of all x. in the ring.

[00:35:18]
So that if you look at the convex hull of x. and k. and. We'll take a break so it's exactly these little hit that I have drawn here we're going to have smaller equal than Delta and one can show that this illumination but these convex and these can also be used to define for general convex bodies defined surface area so that you can look at the limit and stunt or goes to 0 or not the volume difference between that object and dumped into the tool room plus one of these exists and is not some different constant victorie know exactly tense f.n.c. of the serial.

[00:36:03]
Then other evidence that has been explored. Are. Related to these observations by far there at least if the floating body exists then point to the center of gravity of the hyper plane that Delta. And. The center of gravity of these hyping their cuts of South don't have with k. plays no role.

[00:36:33]
So all so then so that would be a guest on point. So then what point can do when looked at the hypo being that cutoff Delta instead of taking the center of gravity of this section what has been done is look at the center of gravity of this camp you so we take the center of gravity of this camp and take also watched interest and that gives us a convex body always and that was done by one who on and on and what cost so $1.00 and was.

[00:37:09]
Paying before us. Shone date these bodies which they called Make sure no eat it troll will eat race convict also now be Staci to call them maybe on so long floating on these. They always can mix and. They also could be used to define or to extent Friends of Syria and there is do a concept of the snow in works with some ph d. students instead of doing these for camps one can of course do the same thing at the center of gravity for a side.

[00:37:56]
So that just to give you a little bit of an overview our outlook or idea. But directions one could if one wanted to go further so am I so paid. So. And that with that I want to finish this chapter on floating bodies and I'll come to all and broke summation of convicts bodies by pretty tops.

[00:38:30]
For the moment. Ok. So very self. A huge amount of literature on netbook summation of context produced by party types with. All kinds of. Conditions cited conditions like approximate can buy a polytope with a fixed number of words from the inside approximating by a fixed a boy put it up with a fixed number affected from the outside or even arbitrary position and all kinds of metrics that tell us how good the approximation is have been considered I have one basically only.

[00:39:54]
Deal mainly deal only with experts that relates these epochs emission business to the previous one which is the floating bodies so approximation. Of economists Bundy's. When you talk to. Them and the 1st point they that want to make really direct relating a question of a book summation to the floating bodies namely I want to present on I agree that cost and called the floating body I agree them.

[00:40:42]
So it was a time when I should and we'll see this will produce a poorly took that approximates who can exploit it so how does the flow rate and goals to let me illustrate it 1st and then so I explain need 1st and then I'll write down the theory so we'll have a coin that.

[00:41:04]
Said case given. And. They do. And the corresponding fluking body of. Delta is 0 so if you can so here you have insight. The. Time. Then what real do next one on the boundary of Ok. So start choosing. X. one on the boundary of b.. Then Oa next step will be a point x. 2 on the boundary of p. with 0 to normal.

[00:41:49]
And next to this ocean all need not be unique so we'll to supporting to x. 2 let me be on the safe side and compute here. So such that. The high population h.t. reassured by the right know what the T.'s so each if the hyper plane goes through the point sorry.

[00:42:19]
So it's the hyper plane that goes through the point x. 2 minus 2 in x. 2 and I was saying a 2nd ago to is and has as l. to normal in x. to. So this so they x. one intersect the interior of. Minus. In t.. Is Here is the x. tool I want to nominate next to.

[00:42:52]
Say in a 2nd but the Delta to use. These type of plane that distance Delta to. 8. 2 and Hubie would have inched minus and. Been interested at the moment so that is how I. And I still have to stay with the Delta tourists so burnt out type 2 is chosen such didn't.

[00:43:24]
Age through my ignorance intersected with the value is equal to. Take so that is how we choose. This 2nd point together with. Distance. Now we keep going. So having TROs and x. one to x. minus one we do x. expected we'll choose. Such they. Say x. one x. k. me or minus one he sent intersects with the interior of age Ok minus interest paid.

[00:44:08]
In t. and such their. Age Minus interest with k. this morning Delta paid so that we keep growing so he respects and so on now the question is of course when does this stop so when when do we stop. In to see the act. Wheel of sort of 2 things so firstly be observed date if we look at leave me Troy here if you look it so what we're looking for or i j that different.

[00:44:54]
We observe if we look at the one that I think an Excel if you'd like so. I x I. Hundreds home with Tom So we'll take this thing here. So we'll take pics. With kiddo top and then real interesting type of plane that is. X. to hit the hyperplane this one h. 2 this x c n x k hyperplane 8.

[00:45:32]
So if we intersect it with 8 minus they test him to introspection if we look at the corresponding thing for. J.. So that's the 1st thing that you observe that stand the constructions was done like so that these works like basically by construction. And then very severe other thing which requires more work namely one or 2 C's cost in order to stay.

[00:46:14]
The value of x. I so for all I. See in. Of course I think. So. So and that requires really work so if we look at x. I expect these little heads that we have drawn here. So they need heads are so yeah so those heads. Intersected with h. and minus this while you roughly don't.

[00:46:59]
But that requires work so then we then we can decide when we stop Namely we have a. So basically what we have is you look at the Union I goes from one so it's to endpoints that we can have people in points that we can choose that things work so and then we look at the volume of the union of these x. i k Delta intersected with age I buy into this this war you is what Because they are disjoint and roughly to doto of the order and times Delta on the other hand these Then why are you these little heads will fill up all the 2.

[00:47:49]
Of caves without a delta so these go on the one hand be there on the other hand it will be off the order came out. And then. How many planes in we have to choose and then we look and then take. P n polytope with these in which is which is the connect tell all of these these x 12 x. and point.

[00:48:20]
Of view it's what takes a month or so we've spoken to here. So they're just roughly the construction that's how the construction goes or the idea a picture of the construction and how am I doing one minute Ok not so good. So. So one hair's. And maybe I can say this rather quickly so one has of course is contained in k. but it is also containing.

[00:48:59]
Maybe I'll say something about this in a 2nd. So basically what is the thing to observe here and to retain is that such an f.o. cremating or such a polytope which gives a broken Mason in some sense and I'll explain next time approximation in which things gives such a thing for you took.

[00:49:20]
The body and a floating point so that's what we want to retain here and that basically also follows by the construction because if we so suppose. K. Delta were not contained in p. n.. Is. Take all poly top. Know it's not complete picture here like. So huge that Delta all probably took would so to speak.

[00:49:54]
Not. Be including that part then we would find an external time in. Such that. Is not in this polytope p. n. But then we can find a hyper plane by separation we can find a hyper plane h. that separates this point from the polytope then what we'll do is we move find separating high plains hyperplane.

[00:50:29]
Plane 8 what we'll do is remove age parallel to an h. bar. Such that the 8. H. bar minus intersected. And then find the same with the same normal a point here x. are on the boundary of k. and their new point we could include in the construction of those and they would.

[00:51:02]
Contradict that we have choice makes him a number of points so that thing works so it's basically contradicting would contradict the mixing of the t.. So we'll have this happening and as I said these people want to keep in mind they can squeeze people see their own abrupt you may well be all you see and know their G.'s and that broke in making polytope but make that more precise next time thank you very much thank you.

[00:51:33]
Yeah. Truth here. So real truth so x. one is. X. one is picked right and then how do you pick the next you choose you so that. If you look at an x. 2 with its ultra normal right. And then the autonomic need not be unique then you'll choose it so that x. one is not in so you will look 8.

[00:52:14]
X. 2 minus. 2. And next to right so you remove these. Type of plane there right so the Delta 2 is such that the hyping of little delta from the body and. You want in addition there x. one is not seen here. Sorry. This is interior. Ball you could be like so like if you picked the x to hear.

[00:53:02]
And then you'll move you to little bits and then you'll get it the x. one India 2 if you pick the x. 2 here right then that must be one then you would be in the interior of the to right. And that's not what you want. To stand for to me this Ok so now I understand what you mean yes yeah so so what i'm not understand what you mean so what is not supposed to happen is these 2 Asian You write so that's not supposed to happen so so like also I do not want the point x. to like to see right.

[00:53:54]
So so basically I should train should not be in on these hyperplane either right. Yes of course you are right yeah now I understand what you mean. That's it clarified Ok. Now. We are actually getting into these when the question young get into this when we when I show you know what's really.

[00:54:28]
Happening while you are in breach sense this is an approximation polytope So IOW show that yeah so there is such a thing indeed. Thank you.