Hardy-type inequalities with sharp constants in domains lambda-close to convex

We justify new integral inequalities with sharp constants for real-valued functions vanishing on the boundary of a domain of Euclidean space on assuming the domain lambda-close to convex. In particular, the closure of such domain is weakly convex in the sense of Efimov–Stechkin and Vial. We describe both standard and strengthen Hardy-type inequalities when instead of the gradients of test functions we use the inner products of the gradients of the distance function from a point to the boundary of the domain by test functions. To prove our main theorem, we apply several lemmas of significance in their own right.