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][1]"
 


  [1]: https://projecteuclid.org/journals/acta-mathematica/volume-91/issue-none/An-isoperimetric-inequality-for-closed-curves-convex-in-even-dimensional/10.1007/BF02393429.full