The Gaussian measure of a convex body controls its maximal covering radius
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913345397850112 |
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| author | Szusterman, Maud |
| author_facet | Szusterman, Maud |
| contents | The well-studied vector balancing constant $β(U, V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $α(U,V)$. In [BS97], Banaszczyk and Szarek proved that $α(B_2^n, V)\leq c$ when $V$ has Gaussian measure at least $\frac{1}{2}$, and conjectured that, for centrally symmetric $V$, $β(B_2^n, V)$ is always bounded by a function of the Gaussian measure of $V$, independent of $n$. We resolve this conjecture in the affirmative. Moreover, we show that the analogous result holds for $α(B_2^n, V)$ even without the central symmetry assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Gaussian measure of a convex body controls its maximal covering radius Szusterman, Maud Metric Geometry Combinatorics The well-studied vector balancing constant $β(U, V)$ of a pair of convex bodies $(U,V)$, is lower bounded by a lattice counterpart, $α(U,V)$. In [BS97], Banaszczyk and Szarek proved that $α(B_2^n, V)\leq c$ when $V$ has Gaussian measure at least $\frac{1}{2}$, and conjectured that, for centrally symmetric $V$, $β(B_2^n, V)$ is always bounded by a function of the Gaussian measure of $V$, independent of $n$. We resolve this conjecture in the affirmative. Moreover, we show that the analogous result holds for $α(B_2^n, V)$ even without the central symmetry assumption. |
| title | The Gaussian measure of a convex body controls its maximal covering radius |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2405.05954 |