Convex order and heat flow for projection profiles of ${\mathrm{\ell }}_p^n$ balls

Let $B_p^n$ be the unit ball of ${\mathrm{\ell }}_p^n$, 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 $t\ge 0$. 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.