Mathematics

A tiling of the unit square is an MTP tiling if the smallest tile can tile all the other tiles. We look at the function f(n)=max∑ s i , where s i is the side length of the i th tile and the sum is taken over all MTP tilings with n tiles. If n= k 2 +3 , it was conjectured that f( k 2 +3)=k+1/k . We show that any tiling that violates the conjecture must consist of at least three tile sizes and has exactly one minimal tile.

We further develop the relationship between β -numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature curv α μ;2 (x,r) at μ - a.e. x∈ R m implies that μ is C 1,α n -rectifiable.

For a metric space X we study metrics on the two copies of X . We define composition of such metrics and show that the equivalence classes of metrics are a semigroup M(X) Our main result is that M(X) is an inverse semigroup, therefore, one can define the C ∗ -algebra of this inverse semigroup. We characterize the metrics that are idempotents, find a minimal projection in M(X) and give examples of metric spaces, for which the semigroup M(X) is commutative. We show that if the Gromov-Hausdorff distance between two metric spaces, X and Y , is finite then M(X) and M(Y) are isomorphic. We also describe the class of metrics determined by subsets of X in terms of the closures of the subsets in the Higson corona of X .

We have shown recently that, given a metric space X , the coarse equivalence classes of metrics on the two copies of X form an inverse semigroup M(X) . Here we give several descriptions of the set E(M(X)) of idempotents of this inverse semigroup and its Stone dual space X ˆ . We also construct σ -additive measures on X ˆ from finitely additive probability measures on X that vanish on bounded subsets.

A convex quadrilateral, Q , is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of Q coincides with the midpoint of at least one of the diagonals of Q . A parallelogram, P, is a special case of a midpoint diagonal quadrilateral since the diagonals of P bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms involving convex quadrilaterals. First, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has tangency chords which are parallel to one of the diagonals of Q . Second, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has a unique pair of conjugate diameters parallel to the diagonals of Q . Finally, we show that there is a unique ellipse, E I , of minimal eccentricity inscribed in a midpoint diagonal quadrilateral, Q , and also that the unique pair of conjugate diameters parallel to the diagonals of Q are the equal conjugate diameters of E I .

We show that the Holmes--Thompson area of every Finsler disk of radius r whose interior geodesics are length-minimizing is at least 6 π r 2 . Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santaló hold on Finsler manifolds with almost no trapped geodesics.

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of O(3) in several cases. We also characterize the convex bodies with the minimal volume product in each case. In particular, this provides a new partial result of the non-symmetric version of Mahler's conjecture in the three dimensional case.

We show that every minimum area isosceles triangle containing a given triangle T shares a side and an angle with T . This proves a conjecture of Nandakumar motivated by a computational problem. We use our result to deduce that for every triangle T , (1) there are at most 3 minimum area isosceles triangles that contain T , and (2) there exists an isosceles triangle containing T whose area is smaller than 2 – √ times the area of T . Both bounds are best possible.

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.

In this paper, the mixed Lp-surface area measures are defined and the mixed Lp Minkowski inequality is obtained consequently. Furthermore, the mixed Lp projection inequality for mixed projection bodies is established.