We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator...
Domingo García, M. Maestre, D. Pinasco et al.· 0 citations
Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely th...
We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-ga...
Daniel Galicer, Mariano Merzbacher, D. Pinasco· 0 citations
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