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Register a new userGeneralized Dissections and Monskys Theorem
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Monskys celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial $p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monskys polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.
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In 1940, Luis Santalo proved a Helly-type theorem for line transversals to boxes in R^d. An analysis of his proof reveals a convexity structure for ascending lines in R^d that is isomorphic to the ordinary notion of convexity in a convex subset of R^{2d-2}. This isomorphism is through a Cremona transformation on the Grassmannian of lines in P^d, which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrary-dimensional flats.
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local dualit
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Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in d-dimensional Euclidean space. Namely, he claimed that there exist at least 2d extremal neighboring spheres. In this paper, we show that there are certain gaps in Schattemans proof, which is based on the Bruggesser-Mani shelling method. We show that using this method it is possible to prove that there are at least d+1 extremal neighboring spheres. However, the existence problem of 2d extremal neighboring spheres is still open.
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