arXiv subjects
Publications and source records attributed to Maud Szusterman.
In the proof of his famous 5K-theorem, W. Banaszczyk introduced a transformation of convex bodies for which the Gaussian measure is monotone. In this note, we present a simplified proof of this monotonicity by slightly modifying Banaszczyk's transform, so that it interacts smoothly with Ehrhard symmetrizations, thereby yielding a somewhat easier proof of the 5K-theorem.
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.
In [SZ], Soprunov and Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all convex bodies in R^n. Together with Saroglou, they proved the characterization among all polytopes [SSZ1] and, for a larger set of inequalities, among all convex bodies [SSZ2]. The conjecture remains open for n \geq 4. In this work, we investigate necessary conditions on the structure of the boundary of a convex body K, for K to satisfy all inequalities. In particular, we obtain a new solution of the 3-dimensional case.
In 2015, I. Soprunov and A. Zvavitch have shown how to use the Bernstein-Khovanskii-Kushnirenko theorem to derive non-negativity of a certain bilinear form $F_Δ$, defined on (pairs of) convex bodies. Together with C. Saroglou, they proved non-negativity of $F_K$ characterizes simplices, among all polytopes. It is conjectured the characterization further holds among all convex bodies. Towards this conjecture, several necessary conditions on $K$ (for non-negativity of $F_K$), were derived. We give a new necessary condition, expressed with isoperimetric ratios, which provides a further step towards a (conjectural) characterization of simplices among a certain subclass of convex bodies.
We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.