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](https://mathoverflow.net/questions/277096/extreme-points-of-set-of-probability-measures-mathcalp-f-int-mathbbr) 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.](https://www.jstor.org/tc/accept?origin=/stable/pdf/3689944.pdf?refreqid=excelsior%3A6c1f2a909f95192f8b30edf1e31ad3de) 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. 



Extreme Points of a set of distributions with moment and/or support constraint
You already know (from the [previous question](https://mathoverflow.net/questions/277096/extreme-points-of-set-of-probability-measures-mathcalp-f-int-mathbbr) on MathOverflow, and in particular [this paper](https://www.jstor.org/stable/3689944)) 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 $P\_1$.


