Small complete caps in $\mathrm{P}\mathrm{G}(4n+1,q)$

In this paper we prove the existence of a complete cap of $\mathrm{P}\mathrm{G}(4n+1,q)$ of size $2(q^{2n+1}-1)\mathrm{/}(q-1)$, for each prime power $q>2$. It is obtained by projecting two disjoint Veronese varieties of $\mathrm{P}\mathrm{G}(2n^2+3n,q)$ from a suitable $(2n^2-n-2)$-dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of $\mathrm{P}\mathrm{G}(4n+1,q)$ is essentially sharp.