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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].

##### MSC:
 14P10 Semialgebraic sets and related spaces 14N05 Projective techniques in algebraic geometry 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
##### Keywords:
convexity; real algebraic variety; projective duality
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