Extension of the sharp fiber estimate to higher codimension

Determine whether the sharp averaged fiberwise L∞ estimate (1.8), with constant ω_n/ω_{n+m}, holds for immersed n-dimensional submanifolds in codimension m>4 and, if so, establish a proof beyond the codimension-four argument.

Background

Theorem 1.2 establishes the sharp averaged fiberwise L∞ estimate for centered ellipsoidal targets in codimensions 1≤m≤4. The authors explain that the proof reduces the normal fibers to four-dimensional convex-geometric estimates, where the spherical Steiner formula and Gauss–Bonnet theorem control the relevant curvature integral. They note that in codimension m≥5 the corresponding boundary has dimension at least three, and no analogous topological identity is available to isolate the mean-curvature term. The flat model shows that the constant ωn/ω{n+m} cannot be increased, so the unresolved issue is whether the same lower bound remains valid in higher codimension.