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From reading the article, I think they mean “determined” in the sense of controlled. They changed the potential distribution in the electrodes, and the single-electron wavefunction changed in a controlled way. The “geometry” is presumably the spatial probability distribution, including the spin.

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Yes and more they know the precise geometry (position) less they know about momentum

... except that in the given context 'geometry' does not mean position.

Yes it does,from the abstract of the original article: "We show that in-plane magnetic-field-assisted spectroscopy allows extraction of the in-plane orientation and full 3D size parameters of the quantum mechanical orbitals of a single electron GaAs lateral quantum dot with subnanometer precision."

That can hardly be interpreted as talking about 'position.' Also, it helps to remember that 1 nanometer is about 10 atom sizes.

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