The distance to the boundary with respect to the Minkowski functional of a polytope

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also explicitly compute the distance function in a collection of examples and observe the new interesting phenomena that arise for such distance functions.

💡 Research Summary

The paper investigates the regularity properties of a distance function defined with respect to the Minkowski functional (gauge) of a convex polytope K in ℝⁿ. For a compact convex set K containing the origin in its interior, the gauge γ_K(x)=inf{λ>0 | x/λ∈K} is a positively 1‑homogeneous, sub‑additive functional that behaves like a norm but may be asymmetric. The authors first recall the dual relationship between γ_K and the support function of the polar set K°: γ_K(x)=h_{K°}(x)=max_{y∈K°}⟨x,y⟩ and γ_{K°}(x)=h_K(x)=max_{y∈K}⟨x,y⟩. From this they derive the basic inequality ⟨x,y⟩≤γ_K(x)γ_{K°}(y) and the representation γ_{K°}(y)=max_{x≠0}⟨x,y⟩/γ_K(x).

A substantial part of the work is devoted to the geometry of the normal cone N(K,x) and the sub‑differential ∂γ_K(x). Lemma 2 shows that for x≠0 the sub‑differential coincides with the intersection of the normal cone at the normalized point x/γ_K(x) and the boundary of the polar set: ∂γ_K(x)=N(K,x/γ_K(x))∩∂K°. At the origin the sub‑differential is the whole polar set, ∂γ_K(0)=K°. Consequently, γ_K is differentiable at a point x iff the sub‑differential is a singleton; in that case the gradient Dγ_K(x) belongs to ∂K° and satisfies ⟨Dγ_K(x),x⟩=γ_K(x). The gauge’s 1‑homogeneity yields the scaling properties ∂γ_K(tx)=∂γ_K(x) and Dγ_K(tx)=Dγ_K(x) whenever the derivative exists.

Lemma 3 establishes a symmetric relationship: v∈∂γ_K(x) ⇔ x∈∂γ_{K°}(v). This duality is used repeatedly to translate statements about γ_K into statements about its polar and vice versa. Lemma 4 treats the case where the normal cone has non‑empty interior (typical for vertices of a polytope). It proves that an open subset of ∂K° is flat and that γ_{K°} is smooth on the interior of the normal cone, with Dγ_{K°}(v)=x for any v in that interior. Lemma 5 shows that if ∂K is of class C^{k,α} near a point, then γ_K inherits the same regularity in a conical neighbourhood of that point.

The authors then specialize to polytopes. A polytope K can be written as the intersection of finitely many half‑spaces ⟨x,v_i⟩≤1, and its polar K° is the convex hull of the outward normals {v_i}. Faces of K correspond to faces of K° via polarity: if z lies in a k‑dimensional face of K, then N(K,z)∩∂K° is an (n−k−1)-dimensional face of K°. Lemma 6 formalizes this correspondence and shows that the normal cone at any boundary point is generated by the outward normals of the facets containing that point. Consequently, γ_K is smooth on the relative interior of each facet, while points on facet intersections (edges, vertices) are singular for γ_K.

With this geometric groundwork, the paper defines the distance to a bounded open set U with respect to γ_K: \