Optimal mean width and metric entropy estimates for convex bodies

What the math gives to ML

The paper establishes an affine-normalization principle: every convex body admits a linear image whose Euclidean covering entropy is controlled, up to universal constants, by that of the crosspolytope, while the optimal mean-width-to-volume-radius ratio is at most O(sqrt(log(en))). The transferable asset is the use of determinant-constrained linear transformations to remove harmful anisotropy without allowing representation collapse, together with mean width and covering entropy as geometric compactness criteria. A practical neural-network adaptation is to insert a learned determinant-one affine preconditioner on hidden representations and optimize a sampled mean-width-to-volume proxy, testing whether it produces more compressible and better-conditioned activations than LayerNorm or whitening.