#### Vol. 12, No. 8, 2018

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The Euclidean distance degree of smooth complex projective varieties

### Paolo Aluffi and Corey Harris

Vol. 12 (2018), No. 8, 2005–2032
##### Abstract

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern–Schwartz–MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of $X$ with the Euler characteristic of an open subset of $X$.

##### Keywords
algebraic optimization, intersection theory, characteristic classes, Chern–Schwartz–MacPherson classes
##### Mathematical Subject Classification 2010
Primary: 14C17
Secondary: 14N10, 57R20
##### Milestones
Received: 3 November 2017
Revised: 3 May 2018
Accepted: 19 June 2018
Published: 4 December 2018
##### Authors
 Paolo Aluffi Department of Mathematics Florida State University Tallahassee, FL United States Corey Harris Max Planck Institute for Mathematics in the Sciences Leipzig Germany