  In this paper we improve the bounds for the Carath\'eodory number, especially
on algebraic varieties and with small gaps (not all monomials are present). We
provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$,
and $[0,1]^n$. We also treat moment problems with small gaps. We find that for
every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such
that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq
d}\rightarrow\mathbb{R}$ which needs at least
$(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$
atoms $l_{x_i}$. Consequences and results for the Hankel matrix and flat
extension are gained. We find that there are moment functionals
$L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\rightarrow\mathbb{R}$ which need to be
extended to the worst case degree $4d$,
$\tilde{L}:\mathbb{R}[x_1,\dots,x_n]_{\leq 4d}\rightarrow\mathbb{R}$, in order
to have a flat extension.
