We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.

In this note we show that if the ratio of the minimal volume V of n-dimensional parallelepipeds containing the unit ball of an n-dimensional real normed space X to the maximal volume v of n-dimensional crosspolytopes inscribed in this ball is equal to n!, then the relation of orthogonality in X is symmetric. Hence we deduce the following properties: (i) if V/v=n! and if n>2, then X is an inner product space; (ii) in every finite-dimensional normed space there exist at least two different Auerbach...

The slicing problem can be reduced to the study of isotropic convex bodies K with $diam\left(K\right)\le c\surd n{L}_K$, where $L_K$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that ${\left|\right|⟨·,\theta ⟩\left|\right|}_{\psi ₂}\le C{L}_K$ for all θ in a subset U of $S^{n-1}$ with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that $ma{x}_{\theta \in S^{n-1}}{\left|\right|⟨·,\theta ⟩\left|\right|}_{\psi ₂}\ge c∜n{L}_K$. In a different direction, we show that good average ψ₂-behaviour of linear functionals on an isotropic...

The purpose of this paper is to continue the investigations on the homothety of unit balls and isoperimetrices in higher-dimensional Minkowski spaces for the Holmes-Thompson measure and the Busemann measure. Moreover, we show a strong relation between affine isoperimetric inequalities and Minkowski geometry by proving some new related inequalities.

Let $(ℝ,|\left|·\right|{\left|}_{\right)}$ be a Minkowski space with a unit ball and let ${\varrho }_H^{}$ be the Hausdorff metric induced by ${\left|\right|·\left|\right|}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to ${\varrho }_H^{Bⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(²,{\varrho }_H^{})$ over any two-dimensional Minkowski space.

We show that, given an n-dimensional normed space X, a sequence of $N={(8/\epsilon )}^{2n}$ independent random vectors ${\left(X_i\right)}_{i=1}^N$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $\Gamma :ℝ\to {ℝ}^N$ defined by $\Gamma x={(⟨x,X_i⟩)}_{i=1}^N$ embeds X in ${\ell }_{\infty }^N$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ${\ell }_{\infty }^N$ with asymptotically best possible relation between N, n, and ε.

We present a general result on regularization of an arbitrary convex body (and more generally a star body), which gives and extends global forms of a number of well known local facts, like the low M*-estimates, large Euclidean sections of finite volume-ratio spaces and others.

We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty }$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset B_E}{(vol\left(B_E\right)/vol\left(\epsilon \right))}^{1/n}\le c_{0}in{f}_{zonoidZ\subset B_F}{(vol\left(B_F\right)/vol\left(Z\right))}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_p$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_p,dimF=n}in{f}_{G\subset L_q,dimG=n}d(F,G){\sim }_{c_{pq}}{n}^{(q/2)(1/q-1/p)}$.

The geometry of random projections of centrally symmetric convex bodies in ${ℝ}^N$ is studied. It is shown that if for such a body K the Euclidean ball $B{₂}^N$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B=P_H\left(K\right)$ is “far” from $P_H\left(B{₂}^N\right)$ then the (random) body B is as “rigid” as its “distance” to $P_H\left(B{₂}^N\right)$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).

Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of $\ell {₁}^{n²}$ is greater than or equal to c√(n/log³n).