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Main discovery The authors introduce a broad class of shapes called convex-like submanifolds in Lorentz–Minkowski space. They attach three geometric objects to each shape: roofs, caps and towers. They prove a volume comparison theorem: if one convex-like submanifold Γ1 lies inside the acausal interior of another Γ2, then the (n−1)-dimensional boundary volume satisfies V_{n−1}(Γ1) ≤ V_{n−1}(Γ2).
How the authors define and study these objects Convex-like is defined by projections. A causal projection is a projection along a timelike or null direction onto a spacelike hyperplane. For any causal projection P, the restriction P|_Γ is a homeomorphism onto the boundary of a bounded convex region in the image hyperplane. A spacelike hyperplane is one whose induced metric is Riemannian (all tangent vectors have positive Lorentzian inner product).
Roofs resemble supporting planes. Towers are one-sided regions bounded by null supports built from functions on the null cone. Caps are the null boundaries of towers and can be seen as non-smooth null hypersurfaces carrying convex-like submanifolds.
Constructions use support functions on the light (null) cones. Given a continuous positively 1-homogeneous function M^+ on the future null cone, the face-up tower Ω^+(M^+) is {x | ⟨x,L⟩ > M^+(L) for all future null L}. A cap is the boundary ∂Ω^+(M^+). For a bounded Γ, the upper and lower support functions are M^±(L)=inf_{p∈Γ}⟨p,L⟩, and the face-up and face-down towers Ω^+(Γ), Ω^−(Γ) intersect to give the acausal interior Ω(Γ), which is nonempty for convex-like Γ.
Key technical facts used - Caps project homeomorphically under any timelike projection: projecting a cap along a timelike direction gives a continuous bijection to the target spacelike hyperplane. - For smooth, compact, connected spacelike submanifolds without boundary (n≥3), there is a local curvature characterization: convex-like is equivalent to certain nonnegativity conditions on the vector-valued second fundamental form and the existence of a spacelike normal field with nonnegative pairings. - Any compact connected spacelike-convex submanifold (the stronger smooth convexity notion) without boundary in dimension n≥3 is convex-like. - Convex-like submanifolds are acausal (no two points are causally related) and uniformly spacelike: there is a finite uniform bound comparing Euclidean and Lorentzian chord lengths on the submanifold. - Nesting is characterized both by projection images and by towers: Γ1 nests in Γ2 if and only if Ω(Γ1) ⊆ Ω(Γ2).
The volume comparison theorem and its meaning The main theorem states: for two convex-like submanifolds Γ1 and Γ2 in R^{n+1}_1, if Γ1 lies in the acausal interior of Γ2, then V_{n−1}(Γ1) ≤ V_{n−1}(Γ2). This generalizes the Euclidean fact that among two nested convex hypersurfaces the outer one has larger boundary volume.
Supported caveats and limitations - The cap method developed here does not extend to pseudo-Euclidean spaces with higher negative index. The authors say the volume comparison for arbitrary negative index requires different geometric deformations and approximations and will appear in a separate work. - Volumes in Lorentz–Minkowski geometry are delicate for non-smooth objects. The authors handle this by constructing bi-Lipschitz maps to round spheres; this makes the volume section technically involved. - The paper records that the three related notions (convex-like, the smooth spacelike-convex condition, and other previously used cocausal conditions) do not imply one another in general. The authors give examples to show these distinctions.
What this adds The paper provides a projection-based notion of convexity adapted to Lorentzian geometry. It supplies geometric tools—roofs, caps and towers—tailored to lightlike directions. And it extends a basic monotonicity of boundary volume from Euclidean convex geometry into the Lorentz–Minkowski setting, subject to the stated technical conditions and limitations.