TY - JOUR
AB - 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 𝜀 ‐net theorem in Banach spaces of type 𝑝>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.
AU - Ivanov, Grigory
ID - 9037
IS - 2
JF - Bulletin of the London Mathematical Society
SN - 00246093
TI - No-dimension Tverberg's theorem and its corollaries in Banach spaces of type p
VL - 53
ER -
TY - JOUR
AB - We study properties of the volume of projections of the n-dimensional
cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3.
We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of $\crosp^n$ onto a k-dimensional subspace for any n>k⩾2.
AU - Ivanov, Grigory
ID - 9098
IS - 5
JF - Discrete Mathematics
SN - 0012365X
TI - On the volume of projections of the cross-polytope
VL - 344
ER -
TY - JOUR
AB - We extend the notion of the minimal volume ellipsoid containing a convex body in Rd to the setting of logarithmically concave functions. We consider a vast class of logarithmically concave functions whose superlevel sets are concentric ellipsoids. For a fixed function from this class, we consider the set of all its “affine” positions. For any log-concave function f on Rd, we consider functions belonging to this set of “affine” positions, and find the one with the minimal integral under the condition that it is pointwise greater than or equal to f. We study the properties of existence and uniqueness of the solution to this problem. For any s∈[0,+∞), we consider the construction dual to the recently defined John s-function (Ivanov and Naszódi in Functional John ellipsoids. arXiv preprint: arXiv:2006.09934, 2020). We prove that such a construction determines a unique function and call it the Löwner s-function of f. We study the Löwner s-functions as s tends to zero and to infinity. Finally, extending the notion of the outer volume ratio, we define the outer integral ratio of a log-concave function and give an asymptotically tight bound on it.
AU - Ivanov, Grigory
AU - Tsiutsiurupa, Igor
ID - 9548
JF - Journal of Geometric Analysis
SN - 10506926
TI - Functional Löwner ellipsoids
ER -