Boyd & Vandenberghe, example 2.13 — what is the "convex set of joint probabilities" in this example? I am reading Boyd & Vandenberghe's Convex Optimization. I can't understand the example below:

*

*what is the "convex set of joint probabilities" in this example?

*what is the convex set $C$ here?


 A: This example from the chapter in which it have been said that convexity of the set is preserved by perspective function. And inverse of this operation give is also a convex set.
p_ij is joint probability of some event u=i and v=j. It's a real number in [0,1]
p let's say is a matrix with size n \times m  which elementwise consist of p_ij. All elements is lie in [0,1] and summ of all elements of matrix is equal to 1.0
convex set of joint probabilites is the set of all possible discrete distribution p. You can show that this is a convex just by definition.
And C is this convex set.
Looks that this example is not very good because all elements of p_ij is divided not into the same number, but number which depends on j.
So this example is bad choice due to specific denominator.
Q: Did you try to write a letter to prof. Stephen Boyd about it?
If use pure algebraic rules then I can suggest the following:
  1.You take some p - it's convex set.


*You take p_ij it's a convex set.

*You evaluate summ for k it's linear combination of p_kj 
 and it's convex set for each k.

*if take 2 and 3 and make cartesian product it's convex set ([2], [3])

*You apply perspective operation to [3] and you receive convex set so 
 f_ij is convex set

*You make cartesian product of all f_ij and receive f distribution.

*Now in fact f will contain tuples which are elementwise contains [0,1]

*Let's intersect this set with unit ball in |f|_1 <= 1
A: *

*to make things clear, let's consider the simple example of $p=(p1, p2)$ in a two dimensional real space, $(x1, x2)$. Then, the convex set of joint probabilities, $p=(p1, p2)$ is the line segment $x1+x2=1, 0 \leq x1, x2 \leq 1$, which is convex of course. Here on this example the convex set of joint probabilities in $mn$ dimensional instead of being $2$ dimensional in our example.

*The convex set $C$ is just the set of probability values I explained above, line segment in $2$ dimensional case.

A: Explanations so far do not do the book justice.
The joint probabilities pij can be seen as n m x n matrix but equivalently as vector of N = m x n nonnegative numbers that sum to one. It helps to picture what is meant if we think of pij as a vector.
As explained in an earlier example 2.5 in the book the set of all possible such probability vectors lie in the probability simplex of affine dimension N-1 embedded in N dimensional space. This simplex is evidently convex, but that is not the point here.
The convex set C in example 2.13 in the book is meant to be any convex subset of this probability simplex.
Each possible vector pij in C then transforms to a vector of conditional probabilities fij of same dimension N by the linear-fractional function given. fij will typically no longer lie in the probability simplex since conditional probabilities need not sum to one.
The claim in the book is that when you sweep out C by varying pij, then fij will sweep out a region f(C) in N-space that is still convex.
