The Convex Hull of a Variety

We present a characterization, in terms of projective biduality, for the hypersurfaces appearing in the boundary of the convex hull of a compact real algebraic variety. 1. Formula for the Algebraic Boundary Convex algebraic geometry is concerned with the algebraic study of convex sets that arise in polynomial optimization. One topic of recent interest is the convex hull conv(C) of a compact algebraic curve C in R. Various authors have studied semidefinite representations [11, 18], facial structure [17, 23], and volume estimates [3, 19] for such convex bodies. In [14] we characterized the boundary geometry of conv(C) when n = 3. The boundary is formed by the edge surface and the tritangent planes, the degrees of which we computed in [14, Theorem 2.1]. Here, we extend our approach to varieties of any dimension in any R. Throughout this paper, we let X denote a compact algebraic variety in R which affinely spans R. We write X̄ for the Zariski closure ofX in complex projective space CP. Later we may add further hypotheses on X , e.g., that the complex variety X̄ be smooth or irreducible. The convex hull P = conv(X) of X is an n-dimensional compact convex semialgebraic subset of R. We are interested in the boundary ∂P of P . Basic results in convexity [10, Chapter 5] and real algebraic geometry [4, Section 2.8] ensure that ∂P is a semialgebraic set of pure dimension n − 1. The singularity structure of this boundary has been studied by S.D. Sedykh [20, 21]. Our object of interest is the algebraic boundary ∂aP , by which we mean the Zariski closure of ∂P in CP. Thus ∂aP is a closed subvariety in CP n of pure dimension n− 1. We represent ∂aP by the polynomial in R[x1, . . . , xn] that vanishes on ∂P . This polynomial is unique up to a multiplicative constant as we require it to be squarefree. Our ultimate goal is to compute the polynomial representing the algebraic boundary ∂aP . We write X for the projectively dual variety to X̄ . The dual variety X lives in the dual projective space (CP). It is the Zariski closure of the set of all hyperplanes that are tangent to X̄ at a regular point. According to the Biduality Theorem of projective geometry, we have (X) = X̄ . We refer to [7, §I.1.3] for a proof of this important result. For any positive integer k we let X [k] denote the Zariski closure in (CP) of the set of all hyperplanes that are tangent to X̄ at k regular points that span a (k−1)-plane. Thus X [1] = X is the dual variety. We consider the following nested chain of algebraic varieties: X [n] ⊆ · · · ⊆ X [2] ⊆ X [1] ⊆ (CP). Our objects of interest is the dual variety, back in CP, to any X [k] appearing in this chain.