 Right, so this is going to be primarily, like, convex stuff, okay, so it's based on joint work with Julius, and Julius will speak about some of the aspects of this work that are happening today. But let's start with, let's say that A in the Minkowski edition of this cool stuff, which is simply defined as a collection of some of course in the theorem of one major theorem. Analysis is the Minkowski theorem of the Minkowski inequality. It's true, it's true for symmetrical sets, but you can still argue that it has to do with the complexity. Interpolation in between them is the family of convex sets, sort of interpolated between N and B, and they'll be sort of the most natural thing is that A, T, B, this is going to be a family of convex sets, and simply by sort of defining this problem A, T. It's not, it's like one dimensional interpolation, but it's not. So the dimensions of the sets are always the same. Higher dimensional interpolation, like this, where I have this theorem, I divide the boundary of the boundary points in RM, it's the family, so it's going to stay that way in Japan continuously. Okay, so now I want to interpolate, so I want the family of convex sets, now parameterized by omega. So how to... Okay, so then there is a natural way to do this. So what you can do then is, they go as a set. So then we get convex set, and then we can let for X in omega, we can let A, X be by construction, of course. So this talk, we will focus the convex interpolation to distinguish it from sort of other interpolation steps. Okay, if we sort of, I mean, we can similarly, as in one dimensional case here, we can sort of apply the Polnikowski inequality and directly see that the volumes of A, X, this is going to be, it's exactly like in the Vanuatu case, because of course very classical, but I want to tell you that this really relies on omega being convex, because if omega is not convex, and you do the convex how, you will not get, typically not get the right sort of sets on the boundary. It doesn't really work unless omega is convex. So what if this talk, so what do we get? Yeah, so we will see. Okay, so again we have a family, a tau, parametrized by the boundary. Okay, and now we're going to sort of get, given a point X in omega, we want to get like a well-defined convex set with definition, so we define the perspectives to the point X. This is the, okay, so for instance, if you have like, if you imagine Brownian motion starting from this point, the harmonic measure will say, so you get the probability of hitting a certain part of the boundary. Okay, so that's the harmonic measure. Okay, so here is an in, it's like a set integral, but, and this is sort of well-defined, and you can sort of think about it. It's just an integral version of the Moschian vision. And one can think about it as, but that's sort of approximating the harmonic measure by a sum of direct, direct measures. And then you get, or if you want, you can go to the, you can go to the support functions. That's maybe a more clean way to do it. Yeah, this is the definition. Okay, and it gives a well-defined interpolation. Okay, and the good thing here is that, this is, I mean, here, omega can be very, very quite general. So, this works for a lot of complex domains. And in question, so far, it gives the same result as the previous one. It's definitely not, it's not the same interpolation. Not the same interpolation. In dimension one, it's the same. But in higher dimensions, it's typically, where you see that it's a complex domain. Yeah, even in a complex domain, this gives a difference. This will be, this will be smaller, but this interpolation will be contained in the complex. So one, okay, so one thing that is different is, of course, it's defined for more. Omega, it's also, I mean, it's also better in a way. So this will be a linear, this is like a linear interpolation, a complex interpolation, which is a non-linear construction. So that makes it sort of simpler. And another thing is that it also preserves some sort of classes of complex sets that are not so preserved by the complex. Okay, interpolation, we have, we saw that, okay, so Brominkowski gave that, now the nth root of the volume is concave. Okay, so another question is, what kind of property do we have in this case? For instance, if I have, say if I have a tau and b tau, a different boundary, if I have two boundary sort of families, then the interpolation of a tau plus b tau is the sum administrative order. Yeah, so it respects Brominkowski's addition and, yeah, Brominkowski's interest. Okay, so then, observation, which says that, okay, but superharmonic, radius, epsilon, we want to put that, we want to put that this is superharmonic. So it means that we want to show people to integrate some volume that's the normalized shear measure. But now, the point here is that if we have the harmonic measure, it has this nice property that the harmonic measure at x is equal to the mean of the harmonic measures of y on this concrete sphere. So mu of x is equal to the mean. But because these interpolations are defined using these harmonic measures, I know how I have that. This is the normalized measure to that unit. So this equality thanks to the definition of the harmonic interpolation implies that a x is the mean. I mean, this would be, we can think of this as approximately if we, okay, so using this and now sort of approximating this sphere measure by a discrete measure. All right, that a x is approximately a sum. This thing is bigger than we want to. Yeah, so it follows basically directly just from that number. I mean, it's the, this is basically that I mean, this is what you would get if you would get the rate. Instead of this is what you would get if instead of ds you would take the corresponding discrete measure. Approximate this with a discrete measure to go to the limit. I mean, you can let them be equal to the sphere measure of the sphere. So take any discrete approximation. So now I want to say that is this sort of, this prompts, this prompts us to introduce a notion of positivity for families of convex sets. So we say that family subpar of an epsilon and ds is contains the mean. Interpolation begins after having families like this that are that sort of subpar. Yeah, yeah, so this is epsilon for small enough. Yeah, but we talk about, I mean, we talk about convex sets could think of. Yeah, it makes more sense to call this subparmonic. Yeah, yeah, yeah. So, yeah. Now clearly then this volume will be again subparmonic because, I mean, if you do this, then the quality goes in the right direction. So here is a base domain. C, convex sets parametrized by set in omega. This is seven. So the total set here sort of at take the product with A times sort of the imaginary parts here. So this now this would be a subset in C and A is wood. Yeah, so A is the union of A set. So here you have A set. So A set is in ORF but then I add sort of this imaginary part. Okay, so now this is a subset in C and but which is sort of independent of the imaginary part. Now the proposition says that well, the family A set is subparmonic in that sense if and only if this subset is pseudo convex. Yeah, yeah, yeah. Yeah, yeah, yeah. Yeah, yeah, yeah. Yeah, yeah, yeah. Yeah, yeah, yeah. Yeah, yeah, yeah. Yeah, yeah. So we have this super convex set like this which is independent of the imaginary part. If you look at the N of the volume of these sort of real parts it's going to be super. And this is like a strengthening of one of those results because it will prove that. I mean it will prove the same thing but with the logarithm of this is super I mean A set That's, I mean, you, it's super complex. And it's very epic, I guess. But Sudevri is the realist, he proves himself that he has a general authority about burglary, but I think he has to know, right? Maybe he would end up with these volumes. That's the thing. No, the thing is, what? Okay, no, the connection with Poo's results is that Poo goes via Tourist of Ammonic Functions. Yeah. And Kiko Pa. Yeah, I mean, he proves this Kiko Pa version for, right, because that's the general thing about burglary. It's a very, yes, using this here would be first. Okay, so the main difference is that Poo uses Prick of Up. I think there was some Tourist of Ammonic Functions, and I guess Julius would talk more about, yeah, convex functions, but here I look at convex assets. And here, but here you get this by just using the original Komekowski thing. But the main point is not really to prove this result is more about understanding the general framework and the, sorry, that's probably that S-box. Yeah, yeah, yeah, he has another result. That is not, that is not connected. You see here, this is really about the convex sets. Yes, yes, and here, I mean, to connect, to connect pseudo-convex sets with convex sets, you really need to use this maximally. I guess, I don't have time. So let me just say, I don't have time to show this. So it's actually a, it's, so the proof here is, so the proof here is not difficult, but there is one, I think, interesting thing when proving that pseudo-convex set like this is sub-harmonic. So this relies on Kieselmann's minimum principle. How do you see that? So, and actually, okay, and Bull's result gave new proof of Kieselmann's minimum principle. So you need to input something like that, but you can also prove Kieselmann's minimum principle by calculation. So, actually, so that is a political sense of this. Yeah, so it's not, yeah, so it's not trivial. It needs some machinery to do it. Okay. Let's say that this function is, but it's not, it's, it's not, yeah, it's not exactly like I was saying. Yeah, yeah, but that, yeah, so you do that. So you go to the support functions, and then the point is you want to, you want to say that the support functions for A set is, you want to say that the support function of A set is super harmonic in set. That's what we need to show, and that follows from Kieselmann's minimum. Basically applied to the leptochrombex indicator.