 an extreme an extreme ray is said to be an extreme ray. So, D is said to be an extreme ray if D is a ray firstly to begin with and the following implication is what is the implication. So, if you can write D in the following form. So, write D as mu D 1 plus mu 1 D 1 plus mu 2 D 2 where mu 1 greater than 0 mu 2 greater than 0 and D 1 D 2 are missed this year. D is said to be an extreme ray of the set S. If D is a ray of S and the following implication is true D is. So, if you can write D as mu 1 D 1 plus mu 2 D 2 where mu 1 is positive mu 2 is positive and D 1 D 2 are rays of S then it has to be it has to be what should be true then D 1 should be equal to some alpha D 2 or some alpha positive. So, when we are talking of rays and extreme rays and so on the problem becomes trivial as soon as one has a bounded set because all these will become 0 on their own. So, these are all discussed only in the context of unbounded convex set. So, then we are talking of so and so consequently that 0 is really not considered a ray unless you know it is for pathological reasons. So, here we are talking of therefore D 1 D 2 etc all nonzero vectors. So, D this vector D is said to be an extreme ray of S if you so if you can if the following implication is true. That means you write it as a conic combination a strict conic combination of two other rays of the set then it has to be that the two rays are actually coincident in one is just a scaled version of the other. So, these are vectors. So, ray is after all is just a vector. So, go back to the definition a vector D is called a ray if the following implication is true that. So, when you are adding the rays you just add them as vectors. So, if you can write if you happen to be able to write D as a conic combination of two rays then it has to be that the two rays are coincident. In that case we say that the ray D is an extreme ray of the set. No in this example I will need to be I will need to show you take a bit of time to determine what the extreme rays are. So, let me but let me show you another example. So, consider so if you just for simplicity let us just take a cone take a cone like this a convex cone like this what are the extreme rays of this set? So, if you take these two any of these vectors you know take this vector with this vector all of these are extreme rays. So, generally one does not talk of a when you are talking of an extreme or an extreme ray or a ray of a set really what you are only referring to is the direction. So, it does not because you can scale a ray and it will remain a ray. So, all you need to really care about is the direction. So, this direction you can say is an extreme ray. So, all points on this direction are the extreme. Why is that the case? No, the cone comprises of all of these points. So, let me ask you a simpler question. First, tell me all the rays of this set before we talk of extreme rays. What are the rays of this set? So, what are the rays of the set? So, the set of all rays of the set is the set itself. All you can because this is a convex cone you add this is closed this is a convex cone you can take conic combinations of any two points in it it will remain in the set. So, you can take any point add to it any a scalar multiple of any other point in the set that is just going to be a conic combination that conic combination will be in this set. So, this for this cone the set of all rays is actually the set itself is this clear? So, now what is this? So, I will just illustrate this for you here is a point x for example, here is my vector D if I have to add D if I add D to x I am effectively going somewhere to just complete this parallel of the field here this is where my point ends up and I can keep doing this I can take any scalar multiple of D and I will still remain in this set. So, effectively I am just translating x along the direction D forever and I will remain in this this will be true for any x that I take and any D that you take within the set. So, the so in this case this is this set S since S is a cone is a convex cone S infinity is actually S set of all research the recession direction set itself. Now, what are the extreme rays that extreme rays are those that if you try to write them as as a strictly positive conic combination of two other rays then then those it cannot be done those two rays have to be those two rays have to be coincident on each other right. So, which means that you if you take a say if you take a say take some point like this this sort of point cannot be an extreme rate this sort of point cannot be an extreme way because I can always think of this as the as a vertex of some parallel of paper like this is a conic combination of this one plus this one or linear combination of these two so some some conic combination of these two for example. So, if I take up so any of a point like this here cannot be an extreme rate. So, if you start thinking about it you will realize that the only extreme rays possible are actually the ones these are extreme rays and these are extreme the word is well chosen they are sort of at an extreme you can imagine them as they solved they they are they are the ones that flank the cone right. Now, this is in R2 or things in RT R3 and higher dimensions become a lot more complicated. So, for example, if I have an ice cream cone right this is an object in R3 what would be the extreme rays of this cone you can see for example, all the ribs of the cone you know all the directions that start from the apex of the cone and go along the shell of it all of those are extreme rays of extreme rays of the cone. So, if I likewise suppose if I if I constructed a cone in R3 which in which I had say which was not an ice cream cone but rather is a hexagonal cone with six sides to it what would be the extreme rays of such a cone be the edges that you would get you know once again the ribs of the cone you know the edges that you would get from that define the hexagon those will be the extreme rays anything along the surface like this which on the phalate surface is not going to be an extreme is not going to be an extreme ray because it would be you can write it always as a combination of two other right two other rays all the rays of the set S can be written as a conic combination of all other rays that answer the answer to that is yes but I will explain in a much in a better way because this becomes a this sort of statement is it anticipates what I want to just say and it is very sharp can be made a very sharp for polyhedron. So, let me come to come to that point now ok. So, what I we were building up basically to what is called the Minkowski while theorem theorem is due to Minkowski and while. So, in the previous class as I was leaving I asked you to imagine what a bounded polyhedron would look like a polyhedron is an is an intersection of half spaces what is a bounded polyhedron how does that look not a polygon so bounded polyhedron would look something like this right it would look like a set like this ok this is for example not an bounded polyhedron what do I mean by it looks like something like this what special about this set you take any point in this set right you can write it as a convex combination of these red corner point or red extreme point right. So, every once you have once a once you have a bounded polyhedron you can just look at its extreme point and you can generate the entire con polyhedron by taking convex combinations of those extreme point ok what what if you have a set like this which is not a bounded polyhedron but an unbounded polyhedron now it is not true anymore that this set has just two extreme point now it is not true that every point can be written as a convex combination of these two red points right there are points here for example which cannot be written as convex combination of these two. So, what is the analogous extension of that earlier observation the observation is that if you have a bounded polyhedron then it is every point can be written as a convex combination of these fixed point these extreme point plus yes so you have so see so what were you saying very nice very good. So, the thing that he has anticipated is actually the statement of the theorem see if you have a point like if you have a polyhedron like this which is unbounded and take a point like this here take this point x this point cannot be written as convex combination of just the extreme point but it can be what he is saying that it can be written as convex combination of the extreme point plus translation with the rays that you have in the set and in fact you can take only it is enough to take only the extreme rays of this ok. So, this is basically the main observation and what that this theorem is that the that this theorem is talking about ok. So, I will just state for state the theorem for you every polyhedron P ok. So, suppose you can polyhedron I am going to represent it in the following form I am going to represent it as x in Rn such that Ax is less than equal to can be represented in the form. So, I need to tell you one blanket assumption that under which this holds and I have not stated here. So, so every polyhedron so we are talking of polyhedra here that have extreme points that has an extreme point. So, every polyhedron that has an extreme point can be represented in the following in the following form. So, the polyhedron is actually the in the following form every point in it is a complex combination of points xk where k ranges over a set capital K and plus a conic combination of points xr where r ranges over a set capital R ok. Now, here the points xk where k belongs to capital K these are these are the extreme points of this of the polyhedron xr where r belongs to R R the extreme rays. So, if you so what this means is you can you just take the convex hull formed by the extreme points of the set look at also the cone generated by the extreme rays of the set and add the two together that actually gives you back your polyhedron this is what this theorem said this is true provided the set has an x has at least one extreme point. So, you take a so equivalent way of writing this equivalently p is equal to the convex hull of the points x1 till sorry points xk where k belongs to convex hull of points xk where k belongs to capital K plus the cone generated by points xr where r belongs to capital R. So, every polyhedron has this sort of can be represented in this kind of form. So, this is very extremely important theorem because it gives us in one shot of view a way of viewing all polyhedron. Every polyhedron is basically a polytope which is a bounded polyhedron created from just the extreme point translated along the cone created by its extreme base. So, you all so to describe a polyhedron completely all you need are the base polytope that is formed by its extreme point and the extreme rays that are there in the polyhedron put and translate one translate the base along the directions defined by the cone and you are done that is that is how you define and describe a polyhedron. This is this is an extremely difficult complicated a powerful theorem where it also seems very intuitive but it is actually very hard to prove it takes quite a few pages to actually do the proof. So, the way as I said you go back to this this sort of this kind of set here this sort of set the way the way this what Minkowski while theorem is saying is that the way this set is generated is that you take the convex hull of the extreme points which is just this segment and keep translating this segment along the two extreme directions the two extreme rays what are the extreme rays in this case this and this are the extreme rays. So, you take this segment translate it along this direction translate it along this direction that will basically generate for you or translate it along the any direction that is that is formed by a linear convex combination of conic combination of these two directions take this direction this direction take conic combinations of those translate it also along those all of that put together will give you the entire entire polyhedron this yeah I am just I mean one would have to calculate this but yeah to me it looks like this ray and this ray would be extreme rays which one if you take a ray below it then it will not be an array of the of the set yeah the one inside is not anything inside here is not going to be an extreme ray which ray are you referring to the one here this one okay let us I can do it but this one this is the if it is inside the if it is in the interior it that is not why so the exact the extreme rays are these the ones that I am calling extreme rays are these directions okay. So, that gives you so that is the basically that is Minkowski Weyl theorem now why is this so important because it actually in one shot gives us also a way of understanding what happens in the first kind of optimization problem with inequality.