Given any aperture less then π, when the dimension of space is large enough, a one-sided cone with such aperture can be fitted in an orthant (hyperoctant).

This could be judged as selfpromotion, as it is the closing remark of my article, but it is relevant to the question. If there are other sources to the fact or related articles, I would feel grateful for any comments.

Given any aperture less then π, when the dimension of space is large enough, a one-sided cone with such aperture can be fitted in an orthant (hyperoctant).}

Actually, it is the contrary. The widest circular cone that can be fitted into an orthant (hyperoctant) has the aperture $$ \varphi=\arccos\sqrt{\frac{n-1}{n}}$$ which approaches zero as n grows to infinity.

This is connected with the other responces that describe how faces of the cube come closer to the middle, but is counterintuitive as one would expect that "there is more room" in higher dimensions. Maybe, it is just that the circular cones capture more of the space, so there is less leftover?

This could be judged as selfpromotion, as it is the closing remark of my article, but it is relevant to the question. If there are other sources to the fact or related articles, I would feel grateful for any comments.