No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p

We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego, California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg theorem, the selection lemma and the weak ‐net theorem in Banach spaces of type >1 . To prove these results, we use the original ideas of Adiprasito, Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon theorem and slightly modified version of the celebrated Maurey lemma

Quantitative Steinitz theorem: A spherical version

Steinitz's theorem states that if the origin belongs to the interior of the convex hull of a set $Q \subset \mathbb{R}^d$, then there are at most $2d$ points $Q^\prime$ of $Q$ whose convex hull contains the origin in the interior. B\'ar\'any, Katchalski and Pach gave a quantitative version whereby the radius of the ball contained in the convex hull of $Q^\prime$ is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version

On the volume of sections of the cube

We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1]n. We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of Rn onto a k-dimensional subspace that maximizes the volume of the intersection. We nd the optimal upper bound on the volume of a planar section of the cube [−1, 1]n , n ≥ 2

Functional John Ellipsoids

We introduce a new way of representing logarithmically concave functions on $\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every $s>0$, we define a class of non-negative functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$. For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to $f$, and we call it the \emph{\jsfunction} of $f$. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John $s$-functions converge to characteristic functions of ellipsoids as $s$ tends to zero and to Gaussian densities as $s$ tends to infinity. As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant $c_d$ multiple of the integral of the pointwise minimum of a properly chosen subfamily of size $3d+2$, where $c_d$ depends only on $d$.Comment: Clean versio