The convex hull of a variety.

*(English)*Zbl 1253.14055
Brändén, Petter (ed.) et al., Notions of positivity and the geometry of polynomials. Dedicated to the memory of Julius Borcea. Basel: Birkhäuser (ISBN 978-3-0348-0141-6/hbk; 978-3-0348-0142-3/ebook). Trends in Mathematics, 331-344 (2011).

This paper deals with the computation of the convex hull of a compact real algebraic variety in \(\mathbb{R}^n\), which affinely spans \({\mathbb R}^n\). The algebraic study of convex sets is of importance in polynomial optimization.

The main result of the paper is the following: For a given \(X\) smooth and compact real algebraic variety that affinely spans \({\mathbb R}^n\), assuming that only finitely many hyperplanes in \(\mathbb{CP}^n\) are tangent to the corresponding projective variety \(X_{\mathbb C}\) at infinitely many points, the algebraic boundary of its convex hull is contained in the union of the dual of \(X^{[k]}\), with \(k=r(X),\dots, n.\)

Here, \(X^{[k]}\) denotes the Zariski closure in \((\mathbb{CP}^n)^*\) – the dual projective space – of the set of all hyperplanes that are tange to \(X_{\mathbb C}\) at \(k\) regular points that span a \((k-1)\)-plane, and \(r(X)\) is the minimal integer \(k\) such that the \(k\)th secant variety of \(X\) has dimension at least \(n-1\).

As the authors show, with this contention of sets one actually can recover the reduced equation of the algebraic boundary of \(X\), as its defining polynomial must be some combination of those defining the \(\{\big(X^{[k]}\big)^*\}_{r(x)\leq k\leq n},\) and one can only consider those varieties having maximal dimension.

The case when \(X\) is not smooth but have only real isolated singularities can also be dealt in a similar fashion. Some computational examples and challenges are presented at the end of the paper.

For the entire collection see [Zbl 1222.00033].

The main result of the paper is the following: For a given \(X\) smooth and compact real algebraic variety that affinely spans \({\mathbb R}^n\), assuming that only finitely many hyperplanes in \(\mathbb{CP}^n\) are tangent to the corresponding projective variety \(X_{\mathbb C}\) at infinitely many points, the algebraic boundary of its convex hull is contained in the union of the dual of \(X^{[k]}\), with \(k=r(X),\dots, n.\)

Here, \(X^{[k]}\) denotes the Zariski closure in \((\mathbb{CP}^n)^*\) – the dual projective space – of the set of all hyperplanes that are tange to \(X_{\mathbb C}\) at \(k\) regular points that span a \((k-1)\)-plane, and \(r(X)\) is the minimal integer \(k\) such that the \(k\)th secant variety of \(X\) has dimension at least \(n-1\).

As the authors show, with this contention of sets one actually can recover the reduced equation of the algebraic boundary of \(X\), as its defining polynomial must be some combination of those defining the \(\{\big(X^{[k]}\big)^*\}_{r(x)\leq k\leq n},\) and one can only consider those varieties having maximal dimension.

The case when \(X\) is not smooth but have only real isolated singularities can also be dealt in a similar fashion. Some computational examples and challenges are presented at the end of the paper.

For the entire collection see [Zbl 1222.00033].

Reviewer: Carlos D’Andrea (Barcelona)