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Autor:
Mussnig, Fabian, Ulivelli, Jacopo
We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saor\'in G
Externí odkaz:
http://arxiv.org/abs/2412.05001
Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given proj
Externí odkaz:
http://arxiv.org/abs/2411.03977
It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton distribution. In case
Externí odkaz:
http://arxiv.org/abs/2410.23720
Autor:
Szarek, Stanislaw, Wolff, Pawel
The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to some norm,
Externí odkaz:
http://arxiv.org/abs/2410.15118
Barthe, Schechtman and Schmuckenschl\"ager proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimi
Externí odkaz:
http://arxiv.org/abs/2410.11460
Autor:
Castrillón, Luca Tanganelli
Given a hyperplane $H$ cutting a bounded, convex body $K$ through its centroid, Gr\"unbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$where $H^+$ is a half-space of boundary $H$. The inequality is sharp and equality is r
Externí odkaz:
http://arxiv.org/abs/2410.12072
Autor:
Iriyeh, Hiroshi, Shibata, Masataka
We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under two kinds of discrete subgroups of $O(3)$ of order four. We also characterize the convex bodies with the minimal volume product in each c
Externí odkaz:
http://arxiv.org/abs/2409.16785
Autor:
Nakamura, Shohei, Tsuji, Hiroshi
Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation an
Externí odkaz:
http://arxiv.org/abs/2409.13611
Autor:
König, Hermann
The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p \ge 10^{15}$
Externí odkaz:
http://arxiv.org/abs/2409.06432
Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santal\'o inequality in its functional form. Inspired by their method, based on the Fokker-Planck sem
Externí odkaz:
http://arxiv.org/abs/2409.05541