Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions
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Yegor Gorodzha, Steven Hoehner, Gil Kur · arXiv: 2610.34651v2 · Summary page
Why cover this story this week?
A ball can be easier to approximate than a cube
Approximating a curved body by a polytope sounds like a straightforward trade between accuracy and complexity. In high dimensions, the choice of complexity measure changes the story. A new preprint gives sharp estimates for the number of vertices needed to approximate a Euclidean ball when error is measured by relative symmetric-difference volume. At fixed error, subexponentially many vertices suffice, whereas the required number of facets grows exponentially with dimension.
The authors extend the vertex result to Gaussian-measured error and show that a cube can require far more vertices than a ball under unrestricted approximation. This overturns the expectation suggested by a classical result for approximations constrained to lie inside the target body. They also sharpen a dimensional constant governing classical smooth-body approximation. For geometric algorithms and learning theory, the lesson is practical as well as conceptual: vertices, facets and containment constraints describe genuinely different approximation problems.
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