Researchers have shown that a smooth immersion of an oriented closed n‑manifold into Euclidean space ⊕^{n+1} encloses a convex set exactly when every ‘shade’ of the manifold is contractible.

A shade is defined for each unit direction u: the set of points where the Gauss map makes an acute angle with u. The new theorem says that if all these shades are contractible, the immersion is a convex embedding. The same conclusion holds if every shade is k‑connected, with the additional requirement that the dimension n does not exceed 2k+1, or that n equals 2k+2 and the homology group H_{k+1}(M;⊕Z) vanishes.

The proof relies on the Chern–Lashof theorem, which connects convexity with total absolute curvature. The authors show that contractible shades force every regular value of the Gauss map to have exactly one preimage; by Chern–Lashof the immersion must then be convex.

The dimension restrictions are sharp. For every k≥0 and n≥2k+3 there exist smooth nonconvex embeddings of the sphere S^n whose shades are still k‑connected. Thus in dimensions higher than 2k+2 convexity cannot be inferred from k‑connected shades alone.

These results resolve a question posed by H. Wente in all dimensions, sharpening previous partial answers.