The Distance Function from a Real Algebraic Variety

For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial ${\mathrm{E}\mathrm{D}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of ${\mathrm{E}\mathrm{D}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely ${\mathrm{E}\mathrm{D}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}_{X,u}(t^2)={\mathrm{E}\mathrm{D}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}_{X^{\vee },u}(q(u)-t^2)$ where $X^{\vee }$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup (X^{\vee }\cap Q{)}^{\vee }$.