A uniform estimate of the relative projection constant

The main goal of the paper is to provide a quantitative lower bound greater than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane. As a consequence, we establish that for every integer $n \geq 4$ there exists an $n$-dimensional normed space $X$ such that for an every hyperplane $Y$ and every projection $P:X \to Y$ the inequality $||P|| > 1 + \left (8 \left ( n + 3 \right )^{5} \right )^{-30(n+3)^2}$ holds. This gives a non-trivial lower bound in a variation of problem proposed by Bosznay and Garay in $1986$.Comment: 16 page

Equilateral dimension of some classes of normed spaces

An equilateral dimension of a normed space is a maximal number of pairwise equidistant points of this space. The aim of this paper is to study the equilateral dimension of certain classes of finite dimensional normed spaces. The well-known conjecture states that the equilateral dimension of any $n$-dimensional normed space is not less than $n+1$. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces, Orlicz-Musielak spaces satisfying an additional condition and every $(n-1)$-dimensional subspace of $\ell^{n}_{\infty}$ we also provide some weaker bounds on the equilateral dimension for every space which is sufficiently close to one of these. This generalizes the result of Swanepoel and Villa concerning the $\ell_p^n$ spaces.Comment: 14 pages, Numerical Functional Analysis and Optimization 35 (2014

Around the Petty theorem on equilateral sets

The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in the normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and N\'emeth about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for every 3 points in the normed plane, forming an equilateral set of the common distance $p$, there exists a fourth point, which is equidistant to the given points with the distance not larger than $p$. We will also improve the example given by Petty and obtain the existence of a smooth and strictly convex norm in $\mathbb{R}^n$, which contain a maximal 4-element equilateral set. This shows that the theorem of Petty cannot be generalized to higher dimensions, even for smooth and strictly convex norms.Comment: 12 pages, 1 figur

Extremal Banach-Mazur distance between a symmetric convex body and an arbitrary convex body on the plane

We prove that if $K, L \subset \mathbb{R}^2$ are convex bodies such that $L$ is symmetric and the Banach-Mazur distance between $K$ and $L$ is equal to $2$, then $K$ is a triangle.Comment: 17 pages, 13 figure

A uniform lower bound on the norms of hyperplane projections of spherical polytopes

Let $K$ be a centrally symmetric spherical polytope, whose vertices form a $\frac{1}{4n}-$net in the unit sphere in $\mathbb{R}^n$. We prove a uniform lower bound on the norms of hyperplane projections $P: X \to X$, where $X$ is the $n$-dimensional normed space with the unit ball $K$. The estimate is given in terms of the determinant function of vertices and faces of $K$. In particular, if $N \geq n^{4n}$ and $K$ is the convex hull of $\{ \pm x_1, \pm x_2, \ldots, \pm x_N \}$, where $x_1, x_2, \ldots, x_N$ are independent random points distributed uniformly in the unit sphere, then every hyperplane projection $P: X \to X$ satisfies the inequality $||P|| \geq 1 + c_nN^{-8n-6}$ (for some explicit constant $c_n$), with the probability at least $1 - \frac{4}{N}.$Comment: 14 page

On the dimension of the set of minimal projections

Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ be its proper linear subspace. The set of all minimal projections from $X$ to $Y$ is a convex subset of the space all linear operators from $X$ to $X$ and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of $X$ and $Y$. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs. Finally, we provide some more refined results in the hyperplane case.Comment: 31 page