# Envelope of Mid-Hyperplanes of a Hypersurface

## TL;DR

This paper investigates the envelope of mid-hyperplanes of a hypersurface, revealing its structure as centers of conics with high-order contact and exploring conditions for smoothness, extending known curve results to higher dimensions.

## Contribution

It extends classical results about mid-hyperplanes from curves to hypersurfaces, characterizing the envelope as conic centers and analyzing smoothness conditions.

## Key findings

- Envelope consists of centers of conics with at least third-order contact.
- Conditions for the envelope to be a smooth hypersurface are provided.
- Counter-example shows properties for curves do not generalize to hypersurfaces.

## Abstract

Given 2 points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers of conics having contact of order at least 3 with the hypersurface at both points. Moreover, we describe general conditions for the EMH to be a smooth hypersurface. These results are extensions of the corresponding well-known results for curves. In the case of curves, if the EMH is contained in a straight line, the curve is necessarily affinely symmetric with respect to the line. We show through a counter-example that this property does not hold for hypersurfaces.