A Quantitative Steinitz Theorem for Plane Triangulations

We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation $G$ with $n$ vertices can be embedded in $\mathbb{R}^2$ in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a $4n^3 \times 8n^5 \times \zeta(n)$ integer grid, where $\zeta(n) \leq (500 n^8)^{\tau(G)}$ and $\tau(G)$ denotes the shedding diameter of $G$, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure

Robust gift wrapping for the three-dimensional convex hull

A conventional gift-wrapping algorithm for constructing the three-dimensional convex hull is revised into a numerically robust one. The proposed algorithm places the highest priority on the topological condition that the boundary of the convex hull should be isomorphic to a sphere, and uses numerical values as lower-prirority information for choosing one among the combinatorially consistent branches. No matter how poor the arithmetic precision may be, the algorithm carries out its task and gives as the output a topologically consistent approximation to the true convex hull

Convex Polytopes: Extremal Constructions and f-Vector Shapes

These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of the cone of f-vectors for 4-polytopes, and thus identifies the existence/construction of 4-polytopes of high ``fatness'' as a key problem. In this direction, the last lecture presents a very recent construction of ``projected products of polygons,'' whose fatness reaches 9-\eps.Comment: 73 pages, large file. Lecture Notes for PCMI Summer Course, Park City, Utah, 2004; revised and slightly updated final version, December 200

Distribution of Aligned Letter Pairs in Optimal Alignments of Random Sequences

Considering the optimal alignment of two i.i.d. random sequences of length $n$, we show that when the scoring function is chosen randomly, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges to a unique limiting distribution as $n$ tends to infinity. This result is interesting because it helps understanding the microscopic path structure of a special type of last passage percolation problem with correlated weights, an area of long-standing open problems. Characterizing the microscopic path structure yields furthermore a robust alternative to optimal alignment scores for testing the relatedness of genetic sequences

Non-crossing frameworks with non-crossing reciprocals

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that if such an embedding is sufficiently generic, then the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation embeddings of Laman circuits. All self-stresses on a planar graph correspond to liftings to piece-wise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure

Diagonality and idempotents with applications to problems in operator theory and frame theory

We prove that a nonzero idempotent is zero-diagonal if and only if it is not a Hilbert-Schmidt perturbation of a projection, along with other useful equivalences. Zero-diagonal operators are those whose diagonal entries are identically zero in some basis. We also prove that any bounded sequence appears as the diagonal of some idempotent operator, thereby providing a characterization of inner products of dual frame pairs in infinite dimensions. Furthermore, we show that any absolutely summable sequence whose sum is a positive integer appears as the diagonal of a finite rank idempotent.Comment: To appear in the Journal of Operator Theor