Makeev [2] stated that every finite Borel measure in $\mathbb R^d$ assigning zero mass to hyperplanes can be cut by $d$ mutually orthogonal hyperplanes so that every pair divides the measure into four equal parts, and outlined a proof strategy, but the key steps were left incomplete. We give a direct and elementary proof. The main ingredient is a Borsuk--Ulam theorem for $O(k)$: every $B_k$-equivariant map from $O(k)$ to a natural representation of the hyperoctahedral group $B_k$ has a zero. An explicit model map has one free orbit of zeros, consisting of the signed eigenbases of a diagonal operator with simple spectrum. A derivative computation and mod-$2$ equivariant degree complete the proof. This is a self-contained proof of a special case of the general Stiefel-manifold theorem in [5].
We construct effective finite group actions on smooth projective complex curves for which Conjecture A of Polishchuk and Van den Bergh fails. For every $h\geq1$, an explicit $S_3$ action on the smooth projective model of $y^2=t^{12h-5}+t$ has quotient of genus $h$ and exactly one ramification orbit, with inertia of ord...
We study the closed spans of finitely many joint orbits of holomorphic multipliers on the Hardy space $H^2(\mathbb D^d)$ of the polydisk. If fewer than $d$ symbols are used, every such span has infinite codimension. The symbols may be unbounded, provided that all orbit vectors belong to the Hardy space. We give two ele...
I. Krishtal, Javad Mashreghi, P. K. Vishwakarma· 0 citations
It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix $A$ in $SL_n(\mathbb{C})$ is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-h...
Gao-Feng Huang, F. Kutzschebauch, Son Nam Tran et al.· 0 citations
We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show t...
Let $M$ be a closed, oriented manifold. The group $\operatorname{Aut}_\bullet(M)$ of based self-equivalences of $M$ is a natural invariant in manifold classification, and Hambleton and Kreck's commutative braid gives a systematic way to compute it for $4$-manifolds. We extend their framework to $BO\langle n+1\rangle$-m...
Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex $(n+1)$-s...
Xi-Nan Dai, W.-D. Deng, Yi-Dong Shi et al.· 0 citations
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