Duality of subanalytic sets

TL;DR

This paper explores a duality concept for subanalytic sets, extending previous work to define an involutive duality that can be iterated, linking hypersurfaces to their tangent sets.

Contribution

It introduces a new, iterative duality framework for subanalytic sets, generalizing prior hypersurface tangent plane dualities and establishing an involution.

Findings

Defined an involutive duality for subanalytic sets.

Extended duality to more general sets beyond hypersurfaces.

Enabled iterative application of the duality procedure.

Abstract

We study the link between a compact hypersurface in $\P^{n+1}$ and the set of all its tangent planes. In this context, we identify $\P^{n+1}$ to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise to a kind of duality which has already been studied Bruce and Romerro-Fuster, and relates a hypersurface to the set of its tangent planes. But in these papers the dual, in this sense, of the set of tangent planes of a hypersurface was not defined and iteration of the procedure was not possible. Therefore we extend this type of duality to more general sets and achieve a procedure which can be iterated and gives in fact an involution.