Related papers: An Optimal Plank Theorem

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the…

Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry…

Let $H$ be a real Hilbert space. In this short note, using some of the properties of bounded linear operators with closed range defined on $H$, certain bounds for a specific convex subset of the solution set of infinite linear…

In this paper we study sharp generalizations of $\dot{F}_p^{0,q}$ multiplier theorem of Mikhlin-H\"ormander type. The class of multipliers that we consider involves Herz spaces $K_u^{s,t}$. Plancherel's theorem proves…

This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…

Piecewise linear vector optimization problems in a locally convex Hausdorff topological vector spaces setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed…

The $n$-dimensional numerical range of a densely defined linear operator $T$ on a complex Hilbert space $\H$ is the set of vectors in $\C^n$ of the form $(< Te_1,e_1>,...,< Te_n,e_n>)$, where $e_1,...,e_n$ is an orthonormal system in $\H$,…

In this work, we consider a class of convex optimization problems in a real Hilbert space that can be solved by performing a single projection, i.e., by projecting an infeasible point onto the feasible set. Our results improve those…

The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown…

Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from magnitudes of linear measurements. In this paper, we assume that the vector is normalized, but…

We prove the Invariant Subspace Conjecture for separable Hilbert spaces.

The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces…

A $(v,k;r)$ Heffter space is a resolvable $(v_r,b_k)$ configuration whose points form a half-set of an abelian group $G$ and whose blocks are all zero-sum in $G$. It was recently proved that there are infinitely many orders $v$ for which,…

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by…

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized vector…

In this paper we solve the polarization problem for real Hilbert spaces, a long-standing conjecture that had remained open for nearly three decades. We also confirm that the only extremal configurations are orthonormal sets. These are…

Let R be the polynomial ring in r variables over a field k, with maximal ideal M and let V denote a vector subspace of the space of degree-j homogeneous elements of R. We study three related algebras determined by V. The first is the…

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator $T$ on a separable infinite-dimensional Hilbert space $H$ such that the orbit $\{T^{n}x;\ n\ge 0\}$ of every non-zero vector $x\in…

We consider a Hilbert space ${\bf H}$ equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity $\Omega^{r}(s,f),\>r\in \mathbb{N}, s>0,$ of…

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between…