I am interested in finding the extreme points of the following set of distributions \begin{align} \mathcal{P}= \left\{F: \int_{\mathbb{R}} |x|^k dF(x)=c \right\} \end{align} where $k,c>0$.

I know that this paper by Winkler is a standard reference on this question.

I was trying to extract the answer from this paper and got that the set of extreme points is given by \begin{align} \mathop{\rm ex}(\mathcal{P}) = \left\{F \in \mathcal{P} : F= (1-t) \delta_{x_1}+t \delta_{x_2}, t\in[0,1], x_1,x_2 \in \mathbb{R} \right\}. \end{align}

However, not sure if this is correct since in this question, for a very similar set, it was pointed out that it must be combination of three mass points instead of two.