Properties of the Level Sets of Some Products of Functions

We are interested about pairs

of

C^2 C 2

-smooth functions

f,g:\mathbb{R}^n\longrightarrow\mathbb{R} f , g : R n ⟶ R

with bounded

{\rm Hess}^+ H e s s +

complements such that their product preserves this property as well. Recall that

{\rm Hess}^+(f) H e s s + ( f )

stands for the set of all points

p\in\mathbb{R}^n p ∈ R n

such that the Hessian matrix

H_p(f) H p ( f )

of the

C^2 C 2

-smooth function

f\colon \mathbb{R}^n\longrightarrow\mathbb{R} f  ⁣ : R n ⟶ R

is positive definite. In this paper we consider two pairs of real-valued functions with empty

{\rm Hess}^+ H e s s +

complements whose products happen to have bounded

{\rm Hess}^+ H e s s +

complements.