We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(\mathcal{T}_X)$ on Fano toric varieties.

Comment: 14 pages, v2: published version, revisions following referee's comments, major changes in Lemma 3.2 and its proof, to appear in EPIGA

• 14J60 - Vector bundles on surfaces and higher-dimensional varieties, and their moduli
• 14M25 - Toric varieties, Newton polyhedra, Okounkov bodies
• 52B20 - Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)