More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ and $K$, $L$ are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.Comment: Minor corrections, some additional references, agnowledgemen

Wulff shapes and a characterization of simplices via a Bezout type inequality

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K).$$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies

Characterization of Simplices via the Bezout Inequality for Mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.Comment: 8 page

Impact of Hepatitis B Exposure on Sustained Virological Response Rates of Highly Viremic Chronic Hepatitis C Patients

Aim. To evaluate the impact of hepatitis B core antibody (anti-HBc) seropositivity in sustained virological response (SVR) rates in treatment-naïve, chronic hepatitis C (CHC) patients with high pretreatment viral load (>800000 IU/mL). Methods. 185 consecutive CHC patients (14.4% cirrhotics, 70.2% prior intravenous drug users) treated with pegylated interferon-a2b plus ribavirin, for 24 or 48 weeks based on viral genotype, were retrospectively analyzed. SVR was confirmed by undetectable serum HCV-RNA six months after the end of treatment schedule. Results. Thirty percent of CHC/HBsAg-negative patients were anti-HBc-positive. Anti-HBc positivity was more prevalent in cirrhotic, compared to noncirrhotic patients (76.9% versus 19.5%, P < .05). Serum HBV-DNA was detected in the minority of anti-HBc-positive patients (1.97%). Overall, 62.1% of patients exhibited SVR, while 28.6% did not; 71.4% of non-SVRs were infected with genotype 1. In the univariate analysis, the anti-HBc positivity was negatively associated with treatment outcome (P = .065). In the multivariate model, only the advanced stage of liver disease (P = .015) and genotype-1 HCV infection (P = .003), but not anti-HBc-status (P = .726), proved to be independent predictors of non-SVR. Conclusion. Serum anti-HBc positivity does not affect the SVR rates in treatment-naïve CHC patients with high pretreatment viral load, receiving the currently approved combination treatment