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Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Moreover, if $n\geq 3$ is even the corresponding convex combination will always contain one of the endpoints $\gamma(0)$ and $\gamma(1)$. And if $n$ is odd then the convex combination will never contain any of these endpoints.

Another example: Closed strictly convex curves exists only in even dimensions. Recall that $\beta :[0,1] \mapsto \mathbb{R}^{n}$ is called strictly convex curve if it has at most $n$ common points with any hyperplane. Schoenberg proved an isoperimetric inequality for such curves "An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces"

Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Moreover, if $n\geq 3$ is even then there are only two possible corresponding convex combinations with $\lfloor \frac{n}{2}\rfloor+1$ points (one containing $\gamma(0)$, and the other one $\gamma(1)$). And if $n$ is odd then there is the unique corresponding convex combination with $\lfloor \frac{n}{2}\rfloor+1$ points (and it never contains any of these endpoints $\gamma(0)$ and $\gamma(1)$).

Another example: Closed strictly convex curves exists only in even dimensions. Recall that $\beta :[0,1] \mapsto \mathbb{R}^{n}$ is called strictly convex curve if it has at most $n$ common points with any hyperplane. Schoenberg proved an isoperimetric inequality for such curves "An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces"

Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Moreover, if $n\geq 3$ is even the corresponding convex combination will always contain one of the endpoints $\gamma(0)$ and $\gamma(1)$. And if $n$ is odd then the convex combination will never contain any of these endpoints.

Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Moreover, if $n\geq 3$ is even the corresponding convex combination will always contain one of the endpoints $\gamma(0)$ and $\gamma(1)$. And if $n$ is odd then the convex combination will never contain any of these endpoints.

Another example: Closed strictly convex curves exists only in even dimensions. Recall that $\beta :[0,1] \mapsto \mathbb{R}^{n}$ is called strictly convex curve if it has at most $n$ common points with any hyperplane. Schoenberg proved an isoperimetric inequality for such curves "An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces"

Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.

Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.

Moreover, if $n\geq 3$ is even the corresponding convex combination will always contain one of the endpoints $\gamma(0)$ and $\gamma(1)$. And if $n$ is odd then the convex combination will never contain any of these endpoints.