What is it about?

Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension 9, at least one vertex is projected onto the section, but for dimension 10 or larger, there are orthogonal projections for which all the vertices are projected outside the section. In fact, this is the case for "most" orthogonal projections, as the dimension tends to infinity.

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Why is it important?

The geometry of finite dimensional normed spaces never ceases to amaze me.

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This page is a summary of: Shadows of Cube Vertices, American Mathematical Monthly, July 2019, Taylor & Francis, DOI: 10.1080/00029890.2019.1606576.
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