  We establish several new inequalities linking classical exponents of
Diophantine approximation associated to a real vector
$\underline{\xi}=(\xi,\xi^{2},\ldots,\xi^{N})$, in various dimensions $N$. We
thereby obtain variants, and partly refinements, of recent results of Badziahin
and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent
as special cases, with new proofs. Similar estimates concerning general real
vectors (not on the Veronese curve) with $\mathbb{Q}$-linearly independent
coordinates are addressed as well. Our method is based on Minkowski's Second
Convex Body Theorem, applied in the framework of parametric geometry of numbers
introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality
result on polar convex bodies.
