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The 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 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 mention (through reference) that convex curves only require $$\lfloor \frac{n+2}{2} \rfloor$$ 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.

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

The 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 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 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.

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

The 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 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 mention (through reference) that convex curves only require $$\lfloor \frac{n+2}{2} \rfloor$$ 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.

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

The 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 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 mention (through reference) that convex curves only require $$\lfloor \frac{n+2}{2} \rfloor$$ 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.

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

The 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 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 $M$?

The 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 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 mention (through reference) that convex curves only require $$\lfloor \frac{n+2}{2} \rfloor$$ 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.

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