Extreme Points of a set of distributions with moment and/or support constraint Let $X$ be a random variable with the distribution $F$ (cdf). 
What are the extreme points of the sets of the form:
\begin{align}
P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P_2&=\left\{ F:   |X| \le d \right\},\\
P_3&=\left\{ F: \int |x|^k dF\le c, \, |X| \le d \right\}.\\
\end{align}
In this question it was shown for the set $P_1$, the set of extrem points are all two mass disributions.  What about $P_2$ and $P_3$? 
It would also be nice if some one can provide a good reference where the subject of finding extreme points of a set of distributions can be found. 
I am familiar with this reference. However, was thinking maybe there is a more modern work or survey on this. 
Edit 1;  Here is the definition of an extrem point:
An extreme point of a convex set, $A$, is a point $x \in A$, with the property
that if $x = ty + (1 − t)z$ with $y,z \in A $and $t \in [0, 1]$, then $y = x$ and/or $z = x$.
Note that all of the sets above are convex. 
 A: You already know (from the previous question on MathOverflow, and in particular this paper) that the extreme points of $P_1$ form a subset of the set that consists of: Dirac measures $\delta_x$ with $|x|^k \leqslant c$, and two-point distributions $F = (1-t) \delta_x + t \delta_y$ with $(1-t) |x|^k + t |y|^k \leqslant c$, $t \in (0, 1)$, $|x|^k < c < |y|^k$. Dirac measures are always extremal. It is easy to see that if $(1-t) |x|^k + t |y|^k < c$, then the corresponding distribution $F = (1-t) \delta_x + t \delta_y$ is not extremal: it is a convex linear combination of $(1-t-\varepsilon) \delta_x + (t+\varepsilon) \delta_y$ and $\delta_x$ for sufficiently small $\varepsilon > 0$. On the other hand, if $(1-t) |x|^k + t |y|^k = c$, then the distribution $F$ is extremal: if it could be written as a convex combination of two distributions $F_1, F_2$, both $F_1, F_2$ would be supported in $\{x, y\}$, and thus their $k$-th moments are necessarily equal to $c$. Thus necessarily $F_1 = F_2 = F$, that is, $F$ is extremal. This characterises the set of extremal points of $P_1$.
Dirac measures $\delta_x$ with $|x| \le d$ obviously form the set of extreme points of $P_2$.
Finally, $P_3 = P_1 \cap P_2$, and if $F \in P_3$ is written as a convex combination of $F_1$ and $F_2$, then necessarily $F_1$ and $F_2$ belong to $P_2$. Therefore, $F \in P_3$ is extremal for $P_3$ if and only if it is extremal for $P_1$.
