Dimension-raising homomorphisms between lattices of convex bodies

作者：Allgaier Luisa; Weisshaupt Heinz^*

来源：Advances in Mathematics, 2016, 289: 212-249.

DOI：10.1016/j.aim.2015.11.016

• Convex bodies
• Dimension raising
• Lattice homomorphisms
• Anti-homomorphisms
• Legendre-Fenchel transform

We settle the first unsolved case of a problem of P.M. Gruber, asked by him in 1991 in [9], namely, to investigate the homomorphisms from the lattice of convex bodies of E-c to the lattice of convex bodies of E-d for c < d. After establishing dimension bounds for such homomorphism in general, we completely describe these homomorphisms for the case d = c + 1, for c >= 3. (The case d = c was settled in [9].) The obtained result is then applied to characterise homomorphisms and anti-homomorphisms from lattices of convex bodies to lattices of convex functions. The relationship between such homomorphisms and anti-homomorphisms is furnished by the Legendre-Fenchel transform.

• 出版日期2016-2-5