Volumes of tubular neighbourhoods of real algebraic varieties

Richard Alexander Wongkew

Vol. 159 (1993), No. 1, 177–184

DOI: 10.2140/pjm.1993.159.177

Abstract

This paper concerns the following problem:

Let V be an algebraic variety in n-dimensional euclidean space. For each pair of positive numbers ρ and R find an upper bound on the volume of the set of points that are within a distance of ρ from V and within a distance R from a fixed point p₀. Obtain this upper bound so that it is independent of the choice of p₀. In particular, does there exist a universal n-th degree polynomial, say P_n(⋅,⋅,⋅) which automatically provides an upper bound upon entering ρ, R and the degree of V ?

Mathematical Subject Classification 2000

Primary: 14P05

Secondary: 51M25

Milestones

Received: 9 December 1989

Revised: 25 November 1991

Published: 1 May 1993