This is a supplement to Vesselin Dimitrov's answer. For $n=2$ the infimum of admissible constants is between $(2/7)^{1/2}\approx 0.5345$ and $8/13\approx 0.6154$, and it has been conjectured that $(2/7)^{1/2}$ is the truth. The lower bound is due to Cassels (J. London Math. Soc. 30, (1955), 119-121), the upper bound is due to [Nowak][1] (Manuscripta Math. 36 (1981/82), 33-46).

**P.S.** Schmidt's book "Diophantine approximation" on page 41 refers to Mack's earlier result, which was slightly improved by Nowak. Schmidt is slightly misleading to give $0.615$ as an upper bound, because Mack's upper bound was $\approx 0.6155$, and Nowak's upper bound is $\approx 0.6154$, both bigger than $0.615$.

  [1]: http://www.digizeitschriften.de/download/PPN365956996_0036/log6.pdf