For n ≥ 2 a construction is given for convex bodies K and L in R n such that the orthogonal projection Ku can be translated inside Lu for every direction u, while the volumes of K and L satisfy Vn(K) > Vn(L). A more general construction is then given for n-dimensional convex bodies K and L such that the orthogonal projection Kξ can be translated inside Lξ for every k-dimensional subspace ξ of R...

We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, those of sections of convex bodies and of sections of their circumscribed cylinders. For L ⊂ R a convex body, we take n random segments in L and consider their ‘Minkowski average’ D. For fixed n, the pth moments of V (D) (1 ≤ p < ∞) are minimized, for V (L) fixed, by the ell...

Let L be a real (Hausdorff) topological vector space. The space %[L] of nonempty compact subsets of L forms a (Hausdorff) topological semivector space with singleton origin when 3C[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L. Then %[L] is locally compact iff L is so. Furthermore, 90.[Z,], the set of nonempty compact convex subsets of L, is the largest ...

Following Bass–Milnor–Serre, we prove that SLn(Z) has the congruence subgroup property for n ≥ 3. This was originally proved by Mennicke and Bass–Lazard–Serre. Let Γn = SLn(Z). The congruence subgroup problem for Γn (solved independently by Mennicke [Me] and Bass–Lazard–Serre [BLS]) seeks to classify all finite-index subgroups of Γn. For l ≥ 2, the level l principal congruence subgroup of Γn, d...

Abstract In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve problem in several classes groups. particular, start with free groups, that is, give an procedure find out if a finitely generated subgroup group is stable. Then further generalize result quasi-convex torsion-free hyperbolic groups limit

Let L := k̄((ǫ)), where k is a finite field with q elements and ǫ is an indeterminate, and let σ be the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σ in L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let f W be the extended affine Weyl group of G. Given x ∈ f W and b ∈ G(L), we have some subgroup of G(L) that...

By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D◦ L of a linear matrix inequality (LMI) L(X) 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called “Perfect” Positivstellensatz. For example, given a generic convex fre...

We prove that the additive group (E∗, τk(E)) of an L∞-Banach space E, with the topology τk(E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E∗, τk(E)) is uniformly homeomorphic to a subset of ` κ 2 for some κ. As an immediat...

This paper will define the random walk on an integer lattice and will approximate the probability that the random walk is at a certain point after a certain number of steps by using a modified version of the Central Limit Theorem. To accomplish this, we will define the characteristic function of the random walk, find the Taylor expansion of this function, and bound the difference between this f...

Characterizing the universal coverings of smooth projective varieties is an old and hard question. Central to the subject is a conjecture of Shafarevich according to which the universal cover X̃ of a smooth projective variety is holomorphically convex, meaning that for every infinite sequence of points without limit points in X̃ there exists a holomorphic function unbounded on this sequence. In t...