The [moment curve](https://en.wikipedia.org/wiki/Moment_curve) is the set of points of the form 
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.


[Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)) tells us that any point in $\overline{M}$ can be expressed as a convex combination of $n+1$ points in $M$. However, when $n=1$ and $n=2$, we can get by with only $n$ points instead of $n+1$ points.


Question: Is this true for all $n$? I.e., can we express any point in $\overline{M}$ as a convex combination of at most $n$ points from $M$?


Edit: In the answer below, eins6180 points out that $n$ points always suffice, answering my question. However, the reference he suggested actually mentions something much stronger. The moment curve is a *convex curve*, that is, a curve in which no $n + 1$ points lie in a single affine hyperplane. [Barany & Karasev](http://arxiv.org/pdf/1112.5942v1.pdf) mention (through reference) that convex curves only require
$$\left\lfloor \frac{n+2}{2} \right\rfloor$$
points to represent any point in their convex hull. The moment curve example is discussed a little bit in this paper by [Holmsen and Roldan-Pensado](http://arxiv.org/pdf/1407.4091.pdf).


I think a dimension-counting argument shows that this bound is tight.



Tighter Caratheodory on the moment curve?
The answer is **yes** for all dimensions.


An old theorem by [Fenchel](http://link.springer.com/article/10.1007/BF01454836) states that for a compact set $K$ in $\mathbb{R}^n$ every point in the convex hull can be either written as a convex combination of at most $n$ points or $K$ can be separated by a hyperplane. Since $M$ is path-connected, such a separation is not possible.


But there are also more modern theorems of this kind available. A good starting point is recent work by [Barany & Karasev](http://arxiv.org/pdf/1112.5942v1.pdf). Their work also implies an affirmative answer to your question.


