CEU Electronic Theses and Dissertations, 2022
| Author | Kalantzopoulos, Pavlos |
|---|---|
| Title | Some log-Concavity in Geometry |
| Summary | We show (with Karoly Boroczky) the well-known log-Brunn-Minkowski conjecture posed by Boroczky, Lutwak, Yang and Zhang \cite{BLYZ12}, for convex bodies which are symmetric with respect to $n$-independent linear hyperplanes. In particular, under this high symmetry we show \begin{equation*} |[h_K^{1- \lambda}h_L^\la mbda]|\geq|K|^{ 1-\lambda}|L|^{ \lambda}, \end{equation*} where $[\cdot]$ stands for the Wulff shape and $h_K, h_L$ are the support functions of $K,L$. Our results strengthen a previous result due to Saroglou \cite{Sar15} (see Bollob\'as-Leader \cite{BoL95}, Uhrin \cite{Uhrin}, Cordero-Erausquin, Fradelizi, Maurey \cite{CEFM04}), treating the unconditional symmetry. We also clarify its equality case and we discuss some consequences including the uniqueness for the solution of the logarithmic Minkowski problem for convex bodies under this symmetry. We characterize (with Karoly Boroczky and Dongmeng Xi) the equality case of the geometric reverse Brascamp-Lieb or Barthe's inequality \cite{Bar98}, that states the following: if $E_i$ be some subspaces in $\R^n$ and $c_i>0$ be some positive numbers that satisfy $\sum_{i=1}^kc_iP_{E_i}=I_n$, then for any non-negative integrable function $f_i:E_i\to[0,\infty)$, $i=1,\ldots,k$, it holds \begin{equation*} \int_{\R^ n}^*\sup_{x=\su m_{i=1}^kc_ix_i ,\, x_i\in E_i}\;\prod_{i= 1}^kf_i(x_i)^{c _i}\,dx \geq \prod_{i=1}^k\l eft(\int_{E_i}f _i\right)^{c_i} . \end{equation*} Here, $P_E$ stands for the orthogonal projection from $\R^n$ onto $E$ and $I_n$ the identity map. It turns out that the extremizers follow almost the same form with the extremizers in the Brascamp-Lieb inequality, found by Valdimarisson \cite{Val08}. However, our argument is quite different from the one used by Valdimarisson \cite{Val08}. We introduce (with Christos Saroglou) a new family of sharp Santal\'o type conjectures, motivated by a recently work of Kolesnikov and Werner \cite{KW}, and we prove them in some cases. For integers $1\leq j\leq k$ denote $s_j$ the elementary symmetric polynomial of $k$ variables and degree $j$ (see \eqref{elemSymPol}). Fix a basis $\{e_m\}$ in $\R^n$ and denote $B^n_j$ the ball of $\ell_j$-norm. We study the following question: If $K_1,\ldots,K_k$ symmetric convex bodies in $\R^n$ that satisfy \begin{equation*} \sum_{l=1 }^ns_j(x_1(l),\ ldots,x_k(l))\l eq {{k}\choose{j}},\qquad \forall x_i\in K_i,\ i=1,\ldots,k, \end{equation*} (where $x(l)$ is the $l$'th coordinate of a vector $x\in\R^n$), then does it hold \begin{equation*} |K_1|\cdots|K_k|\leq |B^n_j|^k \ ? \end{equation*} We were able to prove it in some cases, including the case $j=1$, $j=k$ and the unconditional case for all $j$'s and we set up an equivalent functional form. Our results also strengthen one of the main results in \cite{KW}, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities, excluding the exceptional case $j=1$, can be interpreted as generalizations of the classical Blaschke-Santal\'{o} inequality, which corresponds to the case $j=k=2$. Related, we discuss an analogue of a conjecture due to K. Ball \cite{Ball-conjecture} in the multi-entry setting and establish a connection to the $j$-Santal\'{o} conjecture. |
| Supervisor | Boroczky, Karoly |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2022/kalantzopoulos_pavlo.pdf |
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