 So, let us go to the next. So, this is the definition of a convex hull of a set. Another concept we need is that of a cone. Set S subset which is in Rn is said to be a cone if the following holds. So, what I should what I want to do is I want to take some alpha times S or let me use a different notation let me use lambda. So, take lambda times S what this is essentially is I am taking every vector X in the set and scaling it with lambda, but then I am going to scale it with a non-negative lambda. So, lambda times S is just a set S magnified or shrunk by lambda by a factor lambda. So, every vector X has been just multiplied by a scalar lambda. So, my mistake here, but I am talking of lambda is that are non-negative. So, lambda S is this set which is where every vector has been multiplied by lambda. Now, the set is said to be a cone if lambda S belongs to S or is a subset of S. So, what does this mean belongs to S for all S for all. So, what does this mean? So, you take any X another way of seeing this is for any X S and any lambda greater than equal to 0 the point lambda X also lies in it. Take any point X in S, take any lambda greater than 0 greater than equal to 0 then the point lambda X must lie in it. That is what then the set S is said to be a cone. So, what does a cone look like? So, cone looks like a cone. So, but it does not have to look like a cone. So, for example, a set like this going all the way. So, if I take any vector X here and if I say suppose I scale this guy, this vector it will scale for each here maybe if I scale it by lambda. So, if this is my X, this will be my lambda X or maybe it will get scaled down if my lambda is less than 1, this could be my lambda X. All such points are all in S. So, this sort of set is this is a cone. Any other examples of cones? Line passing through the origin or that is a cone that does not look like a cone. So, this is a line passing through the origin. This is also a cone. Why? Because I can take any point X here, scale it, it lies on the line. Yes. So, now can a cone not pass through the origin? That is not possible because by the definition I am allowed to take lambda equal to 0 here. If I can take lambda equal to 0 then it means that you know if I put lambda equal to 0 then here I am getting the left hand side as just 0. So, 0 must be in this set. So, a cone that is whose apex is lying somewhere other than the origin, this is not a cone, this sort of set. So, I find sets for example are not necessarily cones because they do not have to pass through the origin. But all subspaces, subspaces are cones. So, subspaces are cones, these sort of sets are cones. You do not need to take the full subspace though because you are not multiplying by negative scalars. So, I am not allowing for lambda negative here. So, I am not scaling in the opposite direction here. So, what that means is a set that looks like this say just a ray like this. This is also a cone although this is not clearly not a subspace. Now, is a cone convex? All these examples were convex. What about is a cone always convex? Can you give me an example of a cone that is not convex? I get what you are seeing but probably the word base is not quite the right word. See, what he is saying is base of a cone need not be convex. What you are saying? I can roughly relate to what you are trying to say. Let me give you an example first in R2 and then we can come to this more exotic example. So, what I have drawn here these are axes. This itself is a cone. So, this is my origin I take any point scale it by any scalar lambda greater than equal to 0 that gives me a point on the axis. So, this is a cone I can not only that I can take some other cross like this passing through the origin these two. So, it is union of this line with this line that is also a cone. So, this is also a cone. So, a cone need not be convex. What you are describing is a slightly more interesting sort of object. So, you can it is not easy to do it in R2. In R2 there are limitations. So, I can try to explain to you how it would be. So, just think of some nonconvex shape say for example, some star or something just a cut out of a star on a page shrink it down all the way down to the origin and also grow it all the way till infinity. What you will be carving out in R3 will be this sort of the trace of this star sort of shape that will not be a convex set though because it will have this sort of this kind of boundary. So, that is a cone. So, that is that sort of set is not necessarily a cone is not is a case of a convex set z not a cone. So, suppose if I gave you two cones C1 C2, if I if C1 C2 are cones, then what about what can you say about C1 intersection C2? What about the intersection of two cones? The intersection of two cones is a cone but can also be a trivial cone. See it can happen that your the two cones are like say here is these two lines that they intersect on these two rays like this they are individually cone but the intersection is only the origin. So, the intersection can this is one of the you can have the problem that the intersection of two cones is just is this sort of trivial cone. Now, so we saw that the cone is not necessarily convex. So, this creates this problem of defining analogous things analogous to convex combination. So, if you if I gave you points x1 x2 till xk that belong to say a set S. So, let S be in a set in Rn and I give you points x1 x2 xk belonging to S this then the then this quantity summation alpha i xi let us call this y summation alpha i xi i running from 1 to k where now the alphas do not there is no restriction on their sum but they are just greater than equal to 0. All the alphas are just greater than equal to 0. This is what is called a conic combination. Now, if I take the set of all conic combinations that means the set like this summation alpha i xi i running from 1 to k alphas are greater than equal to 0 x1 till xk belong to S and k is any natural number. So, this is the set of all conic combinations of points in S. This is what we call the conic hull of it or the cone generated by S. Now, can you tell me how is this related to the smallest cone that contains S? How is this related to the smallest cone that contains S? It is so, it is not the intersection of all cones that contain it. It is not the intersection of all cones that contain this and the reason for that is this kind of conic hull is always a convex set. So, this the conic hull is a convex cone that is because of the way we have defined it. You will see it is very easy to see actually that if I take convex combinations of two points like this, two points in the conic hull that convex combination also lies in the conic hull. So, a conic hull is not the smallest cone but the smallest convex cone that contains S. So, just as an example, a very extreme example, suppose I take the set S as just having two points x1, x2 in R2. So, my set is actually these two points, these two points x1, x2. What is the smallest cone containing these two points? It is those two lines or two rays starting from the origin and one passing through x2, one passing through x1. So, the smallest cone containing these two would have been this. So, a point here would not be in the smallest cone containing x1, x2. So, what about the conic hull? What is the conic hull of this x1, x2? When I take combinations alpha1, x1 plus alpha2, x2 where alpha1 and alpha2 are non-negative. What am I generating effectively? Just think of your parallel low pipette or whatever it is called from high school geometry. So, you have x1, x2, alpha1, x1 plus alpha2, x2 would give you. So, this would be x1 plus x2, alpha1, x1 maybe alpha1 would be x1 would be here, alpha2, x2 say here you would get this would be alpha1, x1 plus alpha2, x2, correct. And by doing this when I as I range over alpha1, alpha2 that are non-negative, I will basically span this entire region in. So, the conic hull is usually larger than the smallest cone, but it is the smallest convex cone that contains itself that you can see itself evident here. Smallest convex cone containing x1, x1 and x2 is the V with the you know the interior in it. Excuse me a chance to define hyperplane. So, hyperplane is defined like this. So, you will take a vector A in Rn, take up scalar B in R. A hyperplane is simply all the x's that satisfy A transpose x equals B. So, hyperplane H is this sort of set is x such that A transpose x equals B. You can also write it in the following way, you can also write this as suppose you know that x0 is such that A transpose x0 is equal to B that means it is a point on the it is a point on the hyperplane. And this is the same as writing x such that A transpose x minus x0 equals B sorry x minus x0 equals 0. So, what this means is take any two points x and x0 that lie on the hyperplane where does the vector A point the point the vector A point is orthogonal to the to the segment x minus x0. So, in a hyperplane when you write a hyperplane in this form as x such that A transpose x equals B the vector A is the normal to the hyperplane. So, this is the normal to the hyperplane. So, because the hyperplane can be written in this form x such that A transpose x minus x0 equals 0. Then it that also means that the hyperplane is actually nothing but x0 plus this A perp what is A perp? A perp is that subspace which is perpendicular to A. So, it is the orthogonal complement of A in the entire subspace which is which is which is perpendicular to A. This is the hyperplane is different from what is called a half space. A half space is again let us suppose you have A in Rn B in R and a half space is x such that A transpose x less than equal. So, a half space is a hyperplane and one side of it. This is what we define this sort of set is what is called a half space. So, is a half plane is a hyperplane a convex set? Yes. Is a half space a convex set? Is a half space a convex set? Yes. So, it is it is this hyperplane and all the all of this. So, I can take any two points in it and the segment joining those two points system is always in the half space. So, a half hyperplane and a half and half spaces hyperplanes and half spaces these are always convex. Now, remember I asked you what if I took the wall here was a affine set but the wall and the air in the room all of that together that was not an affine set. So, the wall is an example of that is actually a hyperplane. The wall and all the air on one side of it that is a half space. So, a hyperplane in particular is always affine but half space is not affine. Final concept for today is the is that of a polyhedron. A polyhedron is a polyhedron is the intersection of half spaces intersection of finitely many half spaces. So, if I so what that means is I need to I need vectors like this say a1 till am all belonging to Rn and I need b1 till bm all scalars. Then you look at something like this the x is such that they are common to the m half spaces that are created by these parameters. So, x such that ai transpose x is less than equal to bi for all i. This is a this set P is a polyvedron. Another way to write this is to simply take put all these ai transposes and make them rows of a matrix A stack up the B is into a column vector B into a column vector. We want to pack up the Bis into a column vector B then your polyhedron P can simply be written as x such that Ax is less than equal to b. This for us is a polyhedron. So, is a polyhedron a convex set? Yes, it is the intersection of half spaces that are themselves convex. So, a polyhedron is always convex. A polyhedron is always convex. Is a polyhedron bounded? Is a polyhedron bounded? The answer is no, because a half space is a polyhedron, a hyper plane is a polyhedron. These are not bounded set. A hyper plane is a polyhedron. Can you explain why a hyper plane is a polyhedron? Yeah, I can think of it as an intersection of two half spaces. One where you have, so a hyper plane like this x such that A transpose x equals B is the intersection of x such that A transpose x less than equal to b with x such that A transpose x greater than equal to b. So, it is an intersection of these two half spaces. So, consequently it is also a polyhedron. Hyper plane is a polyhedron. So, every hyper plane is a polyhedron. So, a polyhedron need not be bounded. Now, polyhedron should not be confused with another common term that is used, which is polygon. A polygon often stands for something like this. Some set like this is often what is referred to as a polygon. That is not what we mean by polyhedron. This sort of set is actually, this sort of set, the one I have drawn right here, this is actually not even convex. So, a polyhedron for us is the intersection of half spaces. A polyhedron need not be bounded, but if it is bounded, so a bounded polyhedron is called a polytope. A bounded polyhedron is called a polytope. Now, what I want you to do as an exercise is to think of what a polytope would look like. So, it is an intersection of half spaces such that the overall intersection turns out to be bounded. Bounded means it can be put inside some ball of finite radius. So, but it is itself an intersection of finitely many half spaces. So, we can think about it and we will discuss this next time.