The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in a 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 Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for any three points in a normed plane, forming an equilateral triangle of side p, there exists a fourth point,...

Let $X={\ell }_p$, $p\in (2,+\infty )$. We construct a function $f:X\to ℝ$ which has Lipschitz Fréchet derivative on $X$ but is not a d.c. function.

This article is devoted to an extension of Simons' inequality. As a consequence, having a pointwise converging sequence of functions, we get criteria of uniform convergence of an associated sequence of functions.

A Banach space X has property (E) if every operator from X into c₀ extends to an operator from X** into c₀; X has property (L) if whenever K ⊆ X is limited in X**, then K is limited in X; X has property (G) if whenever K ⊆ X is Grothendieck in X**, then K is Grothendieck in X. In all of these, we consider X as canonically embedded in X**. We study these properties in connection with other geometric properties, such as the Phillips properties, the Gelfand-Phillips and weak Gelfand-Phillips properties,...

C.-M. Cho and W. B. Johnson showed that if a subspace E of ${\ell }_p$, 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with $r^2+s^2<1$, the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of upper p-spaces...

We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite-dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T\in ℬ\left(X_{ius}\right)$ is of the form λI + S with S a strictly singular operator.

We consider the James and Schäffer type constants recently introduced by Takahashi. We prove an equality between James (resp. Schäffer) type constants and the modulus of convexity (resp. smoothness). By using these equalities, we obtain some estimates for the new constants in terms of the James constant. As a result, we improve an inequality between the Zbăganu and James constants.

We use Simonenko quantitative indices of an $𝒩$-function $\Phi$ to estimate two parameters $q_{\Phi }$ and $Q_{\Phi }$ in Orlicz function spaces $L^{\Phi }[0,\infty )$ with Orlicz norm, and get the following inequality: $\frac{B_{\Phi }}{B_{\Phi }-1}\le q_{\Phi }\le Q_{\Phi }\le \frac{A_{\Phi }}{A_{\phi }-1}$, where $A_{\Phi }$ and $B_{\Phi }$ are Simonenko indices. A similar inequality is obtained in $L^{\Phi }[0,1]$ with Orlicz norm.

We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak*-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity...

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in ${ℝ}^n$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of ${ℝ}^n$ satisfying $d(Y\cap K,B_2^k)\le C(1+\surd (k/ln(n/\left(kln(n+1)\right)))$. This formulation of Dvoretzky’s theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.

Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space $E_1$ with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space $E_2$ with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of $E_1$ and $E_2$ can be controlled by the Szlenk index of E, where the Szlenk index...