[Paper Review] Embedding $S^n$ into $R^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$

Vladimir Oliker|ArXiv.org|Jan 15, 2007

Point processes and geometric inequalities14 references6 citations

TL;DR

This paper provides a variational solution to Aleksandrov's problem—constructing a closed convex hypersurface in ℝⁿ⁺¹ with prescribed integral Gauss curvature—by linking it to optimal mass transport on the sphere. The key contribution is a duality between the geometric problem and a Kantorovich-type optimal transport formulation, enabling numerical solutions via linear programming and offering new interpretations of the Gauss map through transport theory.

ABSTRACT

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Motivation & Objective

• To provide a variational proof of existence and uniqueness of a closed convex hypersurface in ℝⁿ⁺¹ with given integral Gauss curvature, including the case of convex polytopes.
• To establish a deep connection between Aleksandrov’s problem and optimal mass transport theory on the sphere.
• To show that the variational formulation of the problem corresponds to a dual problem in optimal transport, with a cost function derived from geometric constraints.
• To enable numerical computation of convex surfaces via linear programming by leveraging the transport-theoretic formulation.
• To provide new geometric interpretations of classical concepts such as the Gauss map through the lens of optimal transport.

Proposed method

• Formulate the problem of prescribing integral Gauss curvature as a variational optimization over radial functions ρ on the unit sphere Sⁿ.
• Define the energy functional 𝒬[ℎ, ρ] involving dual potentials ℎ and ρ, with constraints derived from the Gauss map and support conditions.
• Establish a duality between the geometric variational problem and a Kantorovich optimal transport problem with cost function c(x, N) = ⟨x, N⟩, where x ∈ Sⁿ and N ∈ Sⁿ.
• Use the fact that the Gauss map α_F maps Sⁿ to Sⁿ and induces a pushforward measure μ = σ(α_F(⋅)), which must match the given measure μ.
• Prove that the optimal solution (ℎ̃, ρ̃) of the variational problem corresponds to a transport plan 𝛾̃ ∈ Γ(μ, σ) that maximizes the cost functional 𝒞[𝛾] = ∫ c(x, N) d𝛾(x, N).
• Demonstrate that the duality gap is zero, so the optimal value of the transport problem equals the optimal value of the geometric variational problem.

Experimental results

Research questions

• RQ1 Can Aleksandrov’s problem of constructing a convex hypersurface with prescribed integral Gauss curvature be solved via a variational principle without assuming uniqueness a priori?
• RQ2 How is the problem of prescribing Gauss curvature on Sⁿ related to optimal mass transport on the sphere?
• RQ3 What is the precise form of the cost function and constraints in the optimal transport formulation of the geometric problem?
• RQ4 Can the solution be computed numerically using linear programming techniques based on the transport formulation?
• RQ5 What geometric insights emerge from interpreting the Gauss map and normal vectors through the lens of optimal transport theory?

Key findings

• The paper establishes a variational solution to Aleksandrov’s problem that does not require uniqueness for proving existence, unlike Aleksandrov’s original non-variational approach.
• The solution is shown to be equivalent to a dual optimal transport problem on Sⁿ with cost function c(x, N) = ⟨x, N⟩, where x and N are points on the sphere.
• The optimal transport plan 𝛾̃ corresponds to the Gauss map of the solution hypersurface, and its support lies in the set where ⟨x, N⟩ ≥ 0.
• The duality between the geometric variational problem and the Kantorovich transport problem is exact: the optimal values of both problems are equal.
• The connection allows for numerical computation of convex polytopes from given integral Gauss curvature via linear programming, as the transport problem reduces to a finite-dimensional linear program when μ is atomic.
• The theory provides an 'economics-like' interpretation of the Gauss map: each normal direction N corresponds to a transport of mass from x to N, with the cost ⟨x, N⟩ reflecting geometric alignment.