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For every tuple $d_1,dots, d_lgeq 2,$ let $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}$ denote the tensor product of $mathbb{R}^{d_i},$ $i=1,dots,l.$ Let us denote by $mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in $mathbb{R}^d,$ $d=d_1cdots d_l,$ endowed with the Hausdorff distance, and by $mathcal{B}_otimes(d_1,dots,d_l)$ the subset of $mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of reasonable crossnorms on $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}.$ It is known that $mathcal{B}_otimes(d_1,dots,d_l)$ is a closed, contractible and locally compact subset of $mathcal{B}(d).$ The hyperspace $mathcal{B}_otimes(d_1,dots,d_l)$ is called the space of tensorial bodies. In this work we determine the homeomorphism type of $mathcal{B}_otimes(d_1,dots,d_l).$ We show that even if $mathcal{B}_otimes(d_1,dots,d_l)$ is not convex with respect to the Minkowski sum, it is an Absolute Retract homeomorphic to $mathcal{Q}timesmathbb{R}^p,$ where $mathcal{Q}$ is the Hilbert cube and $p=frac{d_1(d_1+1)+cdots+d_l(d_l+1)}{2}.$ Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to $mathcal{B}_otimes(d_1,dots,d_l)$ is examined.
We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second dual of the
Polynomial spaces associated to a convex body $C$ in $({bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-
Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability g
Let $K$ and $L$ be two convex bodies in $mathbb R^n$, $ngeq 2$, with $Lsubset text{int}, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $lambda$. J. Barker and D. Larman
We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${mathbb R}^n$ with