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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.11451 |
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Table of Contents:
- Let $B_p^n$ be the unit ball of $\ell_p^n$, with $1\le p<2$. We study central densities of one-dimensional marginals of the uniform measure on $B_p^n$ and of its Gaussian heat-flow regularizations. The profile is standardized by multiplying the central density by the standard deviation of the marginal. The key comparison is distributional: if the squared coordinates of one direction majorize those of another, then the corresponding squared projection is larger in convex order. A heat-flow identity turns this distributional comparison into strict Schur convexity of the smoothed central profile at every positive time. Together with the classical central-section theorem at $t=0$, this gives coordinate maximizers and diagonal minimizers for every $t\ge0$. We also evaluate the endpoint constants along the standard coordinate-to-diagonal chain and give a fourth-cumulant criterion for monotonicity of the coordinate profile.