A sausage body is a unique solution for a reverse isoperimetric problem

We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/\lambda$ (a sausage body) is a unique volume minimizer among all $\lambda$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.Comment: 1 figur

Small ball probability for the condition number of random matrices

Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment

The Thermodynamics of Fluid-Phase Benzene via Molecular Simulation

Accurate values for thermodynamic properties throughout the fluid phase are a requirement for the design of separation processes. To date, very few pure substances have been completely characterized because of time and monetary constraints. Low cost computing power now permits complete determination of the thermodynamic properties of pure substances via molecular simulation. Molecular simulation is computational statistical mechanics. Benzene is an important industrial chemical and pharmaceutical precursor. It is the prototypical, symmetric, hexagonal molecule and is an ideal candidate for molecular simulation. The molecular models of three researchers in the field are submitted for Monte Carlo simulation in the virtual laboratories at All claim that their models best represent real benzene. The MC code used for experimentation measures 12 thermodynamic properties with associated errors, and derivatives of the residual Helmholtz energy with respect to density and temperature to order 4. The thermodynamic properties are used to generate a multiparameter fundamental equation of state that represents the model throughout the fluid phase. Thermodynamic properties from the three models are compared to the values from the Goodwin equation of state for benzene. A single model is chosen as the best representative of real benzen

The Thermodynamics of Fluid-Phase Benzene via Molecular Simulation

Accurate values for thermodynamic properties throughout the fluid phase are a requirement for the design of separation processes. To date, very few pure substances have been completely characterized because of time and monetary constraints. Low cost computing power now permits complete determination of the thermodynamic properties of pure substances via molecular simulation. Molecular simulation is computational statistical mechanics. Benzene is an important industrial chemical and pharmaceutical precursor. It is the prototypical, symmetric, hexagonal molecule and is an ideal candidate for molecular simulation. The molecular models of three researchers in the field are submitted for Monte Carlo simulation in the virtual laboratories at All claim that their models best represent real benzene. The MC code used for experimentation measures 12 thermodynamic properties with associated errors, and derivatives of the residual Helmholtz energy with respect to density and temperature to order 4. The thermodynamic properties are used to generate a multiparameter fundamental equation of state that represents the model throughout the fluid phase. Thermodynamic properties from the three models are compared to the values from the Goodwin equation of state for benzene. A single model is chosen as the best representative of real benzen

The production, properties and applications of the zinc imidazolate, ZIF-8.

Venna, Carreon, and Jasinski produced and characterized the first samples of the zinc imidazolate framework ZIF-8 at the University of Louisville in 2010. In this dissertation the production, properties, and applications of this unique metal-organic framework are explored. Previously, only minute laboratory amounts (1/4 gram), of ZIF-8 were produced via time-consuming and expensive processes. Production quantities have been synthesized via both a continuous and a batch process using a spray drying operation to effect separation of the solid product (ZIF-8) from the mother liquor. Approximately 85% of the mother liquor (methanol), can be recovered from the spray dryer resulting in magnitude-of-order savings in time and money. Before any engineering applications could be suggested it was necessary to quantify important physical properties of ZIF-8 not currently available. The density, thermal conductivity, specific heat, and BET surface area were measured via strict ASTM procedures and reported. It was hoped that the massive surface area of ZIF-8 (~ 1300 m2/g), would effect enhanced heat transfer in engineering applications. The Heat Transfer Laboratories at the University of Louisville, served as the testing site for the use of the microparticle ZIF-8 as an agent for enhanced heat transfer when mixed in small vol% in synthetic oil. Unfortunately ZIF-8 delivered no such enhancement

LpL_p-Steiner quermassintegrals

Inspired by an $L_p$ Steiner formula for the $L_p$ affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, $L_p$-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the $L_p$ affine surface areas and the mixed $L_p$ affine surface areas. We investigate the properties of the $L_p$-Steiner quermassintegrals. In particular, we show that they are rotation and reflection invariant valuations on the set of convex bodies with a certain degree of homogeneity. Such valuations seem new and not have been observed before

Random polytopes obtained by matrices with heavy tailed entries

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $\Gamma$). In particular, we show that $$\Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right).$$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic