BISUBDIFFERENTIABILITY OF BICONVEX FUNCTIONS

In this paper we study biconvex functions. We mainly study three types of biconvex functions: lower semicontinuous biconvex functions, canonical biconvex and bisublinear functions. We also consider biconvex functions that are pointwise upper bounds of some nonempty families of 2-affine, biaffine and bilinear functions, respectively. We show that biconvex functions are closely related to convex functions defined on the tensor product of spaces. The restriction to the space 𝑋 × 𝑌 of convex functions defined on the space 𝑋 ⊗ 𝑌 × (𝑋 × 𝑌), the restriction to the space 𝑋 × 𝑌 of convex functions defined on the space 𝑋 ⊗ 𝑌is a biconvex function, the restriction to the space 𝑋 × 𝑌 of sublinear functions defined in 𝑋 ⊗ 𝑌is a bisublinear function. In this paper we study bisubdifferentiability of biconvex functions, canonical biconvex and bisublinear functions and study a number of their properties. In this paper we also study properties of biconjugate functions. In this paper we study the bisubdifferential of the maximum function. Using convex hulls of biconvex functions on the tensor product of spaces, criteria for bisubdifferentiability of biconvex functions are obtained. A charac-terization of biconvex and bisublinear functions is considered. In this paper, a number of properties of the bisubdifferential of bisublinear functions are studied and an analogue of Hörmander's theorem for bisublinear functions is obtained. The product of a seminorm is a bisublinear function. Therefore, the bisubdifferentiability properties of bisublinear functions follow from the property of bisubdiffe-rentiability of a product of seminorms.