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Johannes Hosle

Publications and source records attributed to Johannes Hosle.

7 recordsLinked to original sources
arXiv2026

Monotonicity formulas in positively curved settings with applications to two-phase free boundary problems

Inspired by the monotonicity formula of Alt, Caffarelli, and Friedman, we consider a natural variant of the ACF functional in positively curved 2-dimensional settings where we integrate over sublevel sets of Green's function rather than disks. We prove sharp almost-monotonicity formulas for our new functional in the case of convex planar domains (both with the pole on the boundary and in the interior) and in the setting of 2-dimensional complete manifolds with Euclidean volume growth and nonnegative Gaussian curvature. The tools include the Schwarz-Christoffel formula and Riesz decomposition using conformal coordinates. As a consequence, using quasiconformal estimates, we give a new Lipschitz bound in the manifold setting for minimizers of the two-phase free boundary problem of Alt, Caffarelli, and Friedman, with the constant depending on only the total integral curvature, in contrast to work of Teixeira and Zhang, which gives constants depending on pointwise bounds for the Riemann curvature tensor and its derivatives. Our methods also recover the Lipschitz bound up to a Neumann boundary of Gemmer, Moon, and Raynor in the planar convex domain case.

math.AP↗
arXiv2026

The isomorphic section-projection problem for convex bodies

Let $K, L$ be convex bodies in $\mathbb{R}^n$ with $K$ centered. Assume that $|K \cap θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$. We prove that $|K| \le c\sqrt{n}|L|$, which is sharp up to the choice of the absolute constant. The result gives the sharp isomorphic order in a mixed section-projection comparison problem, complementing the isomorphic Busemann-Petty and Shephard problems. It also removes the John's position assumption from an earlier result of the author, up to an absolute constant factor.

math.MG↗
arXiv2026

Geometric Block Exponents and a Uniform Mixed-Alphabet Sumset Inequality

We study sharp exponents in inequalities for pairs of finite geometric blocks. We characterize exactly when the endpoint $t=1$ determines the optimal exponent and compute the exponent that is uniform in the length of one block. For a two-term first block the answer is $p_0=\log 4/\log 6$. This yields a uniform two-slice max-convolution inequality and, for every $m,d\ge1$, the dimension-free mixed-alphabet sumset bound \[ |A+B|\ge (|A||B|)^{p_0}, \qquad A\subset\{0,1\}^d,\quad B\subset\{0,1,\ldots,m\}^d. \] For every $m\ge2$, the exponent $p_0$ is best possible; for $m=1$, a larger exponent is available.

math.CO↗
arXiv2026

Stolarsky-Type Inequalities in a Max-Convolution Problem

For $m \in \mathbb{N}$, let $q_m := \frac{\log(2m+1)}{2\log(m+1)}$. The max-convolution inequality \begin{align*} \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} &\ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m} \end{align*}for arbitrary sequences $x_0 \ge x_1 \ge ... \ge x_m \ge 0, y_0 \ge y_1 \ge ... \ge y_m \ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova \cite{BDFKK} on the sizes of sumsets in product sets. This inequality was proven for $m = 2$ by Becker, Ivanisvili, Krachun, and Madrid \cite{BIKM} by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x = (1, t, ..., t^{r}, 0, ..., 0)$ and $y = (1, t, ..., t^s, 0, ..., 0)$, $t \in [0, 1]$, for all $m \in \mathbb{N}$ via a comparison of Stolarsky means. Some perturbations are also verified. Finally, we prove the above inequality when one sequence has only two non-zero terms.

math.CO↗
arXiv2020

On the $L_p$-Brunn-Minkowski and dimensional Brunn-Minkowski conjectures for log-concave measures

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when $K\subset L,$ the multiplicative form of the $L_p$-Brunn-Minkowski conjecture holds for Lebesgue measure for $p\geq 1-Cn^{-0.75}$, which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is contained in the other. We also show that the multiplicative version of the $L_p$-Brunn-Minkowski conjecture for the standard Gaussian measure holds in the case of sets containing sufficiently large ball (whose radius depends on $p$). In particular, the Gaussian Log-Brunn-Minkowski conjecture holds when $K$ and $L$ contain $\sqrt{0.5 (n+1)}B_2^n.$ We formulate an a-priori stronger conjecture for log-concave measures, extending both the $L_p$-Brunn-Minkowski conjecture and the Dimensional one, and verify it in the case when the sets are dilates and the measure is Gaussian. We also show that the Log-Brunn-Minkowski conjecture, if verified, would yield this more general family of inequalities. Our results build up on the methods developed by Kolesnikov and Milman as well as Colesanti, Livshyts, Marsiglietti. We furthermore verify that the local version of these conjectures implies the global version in the setting of general measures, and this step uses methods developed recently by Putterman.

math.AP↗
arXiv2018

On the Comparison of Measures of Convex Bodies via Projections and Sections

In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|θ^{\perp}| \le |L \cap θ^{\perp}|$ for all $θ\in S^{n-1}$, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John's position with $|K \cap θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$ then $|K| \le \sqrt{n}|L|$. The condition we consider is weaker than both the conditions $|K \cap θ^{\perp}| \le |L \cap θ^{\perp}|$ and $|K|θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$ that appear in the Busemann-Petty and Shephard problems respectively. Secondly, we appropriately extend the result of Giannopoulos and Koldobsky to various classes of measures possessing concavity properties, including log-concave measures.

math.MG↗