Let ${ℳ}^d$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in ${ℳ}^d$. Let us fix a Lebesgue measure $V_B$ in ${ℳ}^d$ with $V_B\left(B\right)=1$. This measure will play the role of the volume in ${ℳ}^d$. We consider an arbitrary simplex T in ${ℳ}^d$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_B\left(T\right)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_B\left(T\right)$ is zero.

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.