Let $c\_1,\dots, c\_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c\_1,\dots, c\_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a sufficiently large $b$? 


Here is another more involved question: Is there a way we can tell how $b$ scales with $d$ (maybe we can even get a closed form solution)?


A modified version of this problem (related to this question: [Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian](https://mathoverflow.net/questions/229600/moment-matching-construction-of-a-mixture-of-gaussian-distribution-with-lower-m)) is, for what kind of $\delta$, the point $(c\_1,\dots, c\_d+\delta)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a sufficiently large $b$? 



Gaussian and the convex hull of moment curves
Let $P\_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c\_1,\dots, c\_d$ are the same as those of the standard normal distribution $\gamma$. 
By Theorem 3.1 in **[[1](http://www.jstor.org/stable/3689944)]** 
(and the sentence following it there), there is an extreme measure $\nu$ of $P\_d$. By Theorem 2.1 and Example 2.1 (a) in **[[1](http://www.jstor.org/stable/3689944)]** or Corollary 11 in **[[2](http://link.springer.com/article/10.1007/s00186-015-0530-0)]**, 
$\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c\_1,\dots, c\_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$. 


This provides a positive answer to the question if the $[0,b]$ is replaced by $[-b,b]$. Actually, the answer will remain positive for any probability measure on $\mathbb R$ in place of the standard normal distribution $\gamma$. 


The answer to the original question is an obvious no, since for any random variable $X$ with support set contained in $[0,b]$ and $EX^2=\int\_{\mathbb R}x^2\nu(dx)=1>0$, one would have $EX>0=\int\_{\mathbb R}x\nu(dx)$. 


