Volume of mixed bodies

By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body, These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body, The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies, As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.

A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its i i th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.

About a decade ago Lutwak, Yang, and Zhang introduced the notion of $L_p$-projection body. More recently, Wang and Leng established an $L_p$-version of Petty's affine projection inequality. At the same time Ludwig discovered a family of general $L_p$-projection bodies and Haberl and Schuster established Petty's projection inequality for general $L_p$-projection bodies. In this paper we establish a general $L_p$-version of Petty's affine projection inequality for general $L_p$-projection bodies. Moreover, we obtain an analogous inequality for $L_p$-geominimal surface area.

Affine vs. Euclidean isoperimetric inequalities

Using results of K. Kiener and the Riesz-Sobolev convolution inequality we give a new proof of Petty's projection inequality. By the same method we also obtain a proof of Santalo's affine isoperimetric inequality.

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the "dual" -Petty projection inequality, and the "dual" -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.

For every convex body $$K \subset \mathbb {R}^n$$ K ⊂ R n and $$\delta \in (0,1)$$ δ ∈ ( 0 , 1 ) , the $$\delta $$ δ -convolution body of K is the set of $$x \in \mathbb {R}^n$$ x ∈ R n for which $${\left| K \cap (K+x)\right| }_{n}\ge \delta {\left| K\right| }_{n}.$$ K ∩ ( K + x ) n ≥ δ K n . We show that for $$n=2$$ n = 2 and any $$\delta \in (0,1)$$ δ ∈ ( 0 , 1 ) , ellipsoids do not maximize the volume of the $$\delta $$ δ -convolution body of K, when K runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $$\delta \rightarrow 1^-$$ δ → 1 - , which is governed by the Petty projection inequality.

Recently, Lutwak, Yang and Zhang posed the notion of L p -projection body and established the L p -analog of the Petty projection inequality. In this paper, the notion of L p -mixed projection body is introduced—the L p -projection body being a special case. The Petty projection inequality, as well as Lutwak's quermassintegrals (L p -mixed quermassintegrals) extension of the Petty projection inequality, is established for L p -mixed projection body.

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in Rn from those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm from those in Rn (or vice versa). In this work, we treat the Lp extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m-dimensional convex bodies containing the origin. We establish mth-order Lp isoperimetric inequalities, including the mth-order versions of the Lp Petty projection inequality, Lp Busemann-Petty centroid inequality, Lp Santaló inequalities, and Lp affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn,‖⋅‖E)→(Rm,‖⋅‖F).

The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter‐dimensional operator, Petty's inequality was generalized in Haddad, Langharst, Putterman, Roysdon, and Ye to the so‐called setting, where is an ‐dimensional compact convex set. In this work, we further extend the Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with ‐concave density (for ). Moreover, when , and motivated by a contemporary empirical reinterpretation of Petty's result by Paouris, Pivovarov, and Tatarko, we explore empirical analogues of this inequality.

Study was conducted in the regency of Minahasa to estimate horse live weight using its chest girth, body length and body volume formula (cylinder volume formula) represented by animal chest girth and body length dimensions, particularly focused in Minahasa local horses. Data on animal live weight (LW), body length (BL), chest girth (CG) and body volume were collected from 221 stallions kept by traditional household farmers. Animal body volume was calculated using cylinder volume formula with CG and BL as the components of its formula. Regression analysis was carried out for LW with all the linear body measurements. The data were classified on the basis of age. Age significantly (P < 0.05) influenced the body measurements except animal body length (P>0.05). Animal live weight was predicted by simple regression models using dependent variable (Y) of the animal live weight and independent variable (X) of the animal body measurement, either body length, chest girth, or body volume. The correlations between all pairs of measurements were highly significant (P < 0.01) for all age groups. Regression analysis showed that live weight could be predicted accurately from body volume (R2= 0.92) and chest girth (R2= 0.90). Simple regression model that can be recommended to predict horse live weight based on body volume with their age groups ranging from 3 to ≥10 years old was as follow: Live weight (kg)= 5.044 + 1.87088 body volume (liters). The analyses of data on horse chest girth, body length and body volume formula provided quantitative measure of body size and shape that were desirable, as they enable genetic parameters for these traits to be estimated and also included in breeding programs.

Hystrix javanica is endemic species in Indonesia. Study about fetal development of Hystrix javanica are very rare because of sample limitation. This study was carried out to describe the morphometrics and x-ray analysis of three fetuses in different stage to give basic information about fetal development of Hystrix javanica. Three fetus samples fixed in Bouin’s solution was used in this study. Observation was carried out to identify the characteristic of three fetus samples. This included the pattern of hair, body measurements, body volume, and body weight. X-ray analysis was carried out to know the ossification process in the fetal development. Statistical analysis was carried out using Microsoft 365® Excel program software. Three fetus samples had different specific hair pattern, that was hairless, smooth hairs, and smooth hairs with dense-non dense pattern. Body volume of 1st, 2nd, and 3rd fetus were 23mL, 90mL, and 170mL, respectively. Body weight of 1st, 2nd, and 3rd fetus were 19.5g, 79.22g, and 153.18g, respectively. Pearson’s correlation analysis shown strong relationship between total body length, front body length, back body length, horizontal body diameter, vertical body diameter, head length, and head diameter against body volume and body weight of three fetuses. Significant positive correlation was shown between horizontal body diameter, vertical body diameter, and head diameter against body volume and body length with P value < 0.05. Faint radiopaque images showed in the 2nd fetus sample and strong radiopaque images showed in the 3rd fetus sample. Radiopaque images were identified in the teeth, cranium, vertebrae, and extremities bones. In this study we concluded that there was a specific hair pattern in different fetal stage. All body measurements have positive correlation with body volume and body weight and x-ray analysis shown that the ossification of the bone was started to happen while the smooth hair was growth.

This study aims to develop and validate a model for predicting the body weight (BW) of Ongole Crossbred (OC) cattle using body measurements. To achieve this, a combination of meta-analysis and field experiments was employed. The meta-analysis involved identifying relevant keywords and databases, reviewing titles and abstracts, extracting data, and subsequently tabulating and analyzing the data. A total of 1,141 animal records were included in the quantitative synthesis process. Following the meta-analysis, a BW prediction model for OC cattle was developed. The model incorporated recommendations obtained from the meta-analysis, considering body measurement, age, and sex. Data from 507 animals were utilised to construct the model. Finally, a field experiment was conducted on 35 animals to assess the accuracy of the model. The meta-analysis revealed that body volume (BV) (r=0.96) and heart girth (HG) (r=0.89) exhibited stronger correlations with BW compared to body length (BL) (r=0.68). Linear regression modeling of OC cattle BW, demonstrated that HG yielded high correlation coefficients for both male (r=0.98) and female (r=0.94) cattle. Similarly, BV showed strong correlations for male (r=0.99) and female (r=0.95) cattle. Furthermore, the analysis revealed that both HG and BV were effective predictors across different age groups, with high correlation coefficients observed for cattle aged 1-12 months and over 24 months. The field experiment confirmed the high reliability of the model, achieving an accuracy of 90.8% for HG and 91% for BV. In conclusion, HG and BV are strong predictors of OC cattle BW, with categorization by breed further improving prediction accuracy.

Objective:The aim of the study was to use a meta-analysis to identify the correlation between linear body measurements, including body length (BL), wither height (WH), heart girth (HG), and body volume (BV), and body weight in beef cattle by breed, sex, and age as categories.Materials and methods:These results can be used as a method for predicting beef cattle body weight. This study used systematic review and meta-analysis guidelines to create a checklist. The first stage was searching for papers relevant to the study objectives. The second stage was searching using the keywords beef cattle, body weight, body measurement, and correlation. The third stage was reviewing the title and abstract. The fourth stage was abstracting information from selected papers, and the last stage was tabulating data.Results:The results from this study were obtained, and 32 papers were eligible for the meta-analysis stage. The correlation between linear body measurement and body weight of beef cattle showed that HG (r = 0.88) and BV (r = 0.97) were significantly (p < 0.05) different compared to BL (r = 0.74) and WH (r = 0.72). The correlation between HG and body weight, and the categorization of cattle breeds showed significantly (p < 0.05) different results. The correlation between BV and body weight of cattle according to breed categories showed results that were not significantly (p > 0.05) different, while age was significantly (p < 0.05).Conclusion:In conclusion, to predict beef cattle body weight, it is necessary to use HG or BV, with breed, sex, and age of cattle as categories.

The research examined the efficacy of regional volumes of thigh ratios assessed by stereovision body imaging (SBI) as a predictor of visceral adipose tissue measured by magnetic resonance imaging (MRI). Body measurements obtained via SBI also were utilized to explore disparities of body size and shape in men and women. One hundred twenty-one participants were measured for total/regional body volumes and ratios via SBI and abdominal subcutaneous and visceral adipose tissue areas by MRI. Thigh to torso and thigh to abdomen-hip volume ratios were the most reliable parameters to predict the accumulation of visceral adipose tissue depots compared to other body measurements. Thigh volume in relation to torso [odds ratios (OR) 0.44] and abdomen-hip (OR 0.41) volumes were negatively associated with increased risks of greater visceral adipose tissue depots, even after controlling for age, gender, and body mass index (BMI). Irrespective of BMI classification, men exhibited greater total body (80.95L vs. 72.41L), torso (39.26L vs. 34.13L), and abdomen-hip (29.01L vs. 25.85L) volumes than women. Women had higher thigh volumes (4.93L vs. 3.99L) and lower-body volume ratios [thigh to total body (0.07 vs. 0.05), thigh to torso (0.15 vs. 0.11), and thigh to abdomen-hip (0.20 vs. 0.15); P < 0.05]. The unique parameters of the volumes of thigh in relation to torso and abdomen-hip, by SBI were highly effective in predicting visceral adipose tissue deposition. The SBI provided an efficient method for determining body size and shape in men and women via total and regional body volumes and ratios. Am. J. Hum. Biol. 27:445-457, 2015. © 2015 Wiley Periodicals, Inc.