The maximum number of faces of the Minkowski sum of two convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two $d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number of vertices of the polytopes. For even dimensions $d\ge{}2$, the maximum values are attained when $P_1$ and $P_2$ are cyclic $d$-polytopes with disjoint vertex sets. For odd dimensions $d\ge{}3$, the maximum values are attained when $P_1$ and $P_2$ are $\lfloor\frac{d}{2}\rfloor$-neighborly $d$-polytopes, whose vertex sets are chosen appropriately from two distinct $d$-dimensional moment-like curves.Comment: 37 pages, 8 figures, conference version to appear at SODA 2012; v2: fixed typos, made stylistic changes, added figure

Log-concavity and lower bounds for arithmetic circuits

One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum\_{i = 0}^d a\_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau a\_{i-1}a\_{i+1} for every $i \in \{1,\ldots,d-1\},$ where \tau \textgreater{} 0. Whenever $f$ can be written under the form $f = \sum\_{i = 1}^k \prod\_{j = 1}^m f\_{i,j}$ where the polynomials $f\_{i,j}$ have at most $t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f\_{i,j}$ have only non-negative coefficients, we improve this degree bound to $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if \tau \textgreater{} 1, and to $d \leq kmt$ if $\tau = d^{2d}$. This investigation has a complexity-theoretic motivation: we show that a suitable strengthening of the above results would imply a separation of the algebraic complexity classes VP and VNP. As they currently stand, these results are strong enough to provide a new example of a family of polynomials in VNP which cannot be computed by monotone arithmetic circuits of polynomial size

The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$ in $\reals^d$, as a function of the number of vertices of the polytopes, for any $d\ge{}2$. Expressing the Minkowski sum as a section of the Cayley polytope $\mathcal{C}$ of its summands, counting the $k$-faces of $P_1+P_2+P_3$ reduces to counting the $(k+2)$-faces of $\mathcal{C}$ which meet the vertex sets of the three polytopes. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes in $\reals^d$, where $r\ge d$. For $d\ge{}4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves

Almost simplicial polytopes: the lower and upper bound theorems

International audiencethis is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d

A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ and $r<d$, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [2]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as $f$- and $h$-vector calculus and shellings, and generalizes the methodology used in [15] and [14] for proving upper bounds on the $f$-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum $P_1+...+P_r$ as a section of the Cayley polytope $\mathcal{C}$ of the summands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding the subset of the $(k+r-1)$-faces of $\mathcal{C}$ that contain vertices from each of the $r$ polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.Comment: 43 pages; minor changes (mostly typos