• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.139-148
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• 2018

In this paper, we prove a stability result for p-centroid bodies with respect to the Hausdorff distance. As its application, we show that the symmetric convex body is determined by its p-centroid body.

• East Asian mathematical journal
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• v.16 no.2
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• pp.363-370
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• 2000

We evaluate the dual quermassintegrals of $\gamma$-equichordal bodies and obtain the lower bound for the volume product of p-centroid body in $E^2$.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.403-408
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• 2012

In this paper, we prove that if K is a convex body in $E^n$ and $E_i$ and $E_o$ are inscribed ellipsoid and circumscribed ellipsoid of K respectively with ${\alpha}E_i=E_o$, then $\[({\alpha})^{\frac{n}{p}+1}\]^n{\omega}^2_n{\geq}V(K)V({\Gamma}^{\ast}_pK){\geq}\[(\frac{1}{\alpha})^{\frac{n}{p}+1}\]^n{\omega}^2_n$. Lutwak and Zhang[6] proved that if K is a convex body, ${\omega}^2_n=V(K)V({\Gamma}_pK)$ if and only if K is an ellipsoid. Our inequality provides very elementary proof for their result and this in turn gives a lower bound of the volume product for the sets of constant width.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.465-479
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• 2012

For $0<p{\leq}{\infty}$ and a convex body $K$ in $\mathbb{R}^n$, Lutwak, Yang and Zhang defined the concept of dual $L_p$-centroid body ${\Gamma}_{-p}K$ and $L_p$-John ellipsoid $E_pK$. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $K$, there exist an ellipsoid $E$ and a parallelotope $P$ such that for $1{\leq}p{\leq}2$ and $0<q{\leq}{\infty}$, $E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$; For $2{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, $2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP$ and $V(E)=V(K)=V(P)$. (ii) For any convex body $K$ whose John point is at the origin, there exists a simplex $T$ such that for $1{\leq}p{\leq}{\infty}$ and $0<q{\leq}{\infty}$, ${\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT$ and $V(K)=V(T)$.

• Imaging Science in Dentistry
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• v.51 no.2
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• pp.167-174
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• 2021

Purpose: Sex determination can be done by morphological analysis of different parts of the body. The mastoid region, with its anatomical location at the skull base, is ideal for sex identification. Statistical shape analysis provides a simultaneous comparison of geometric information on different shapes in terms of size and shape features. This study aimed to investigate the geometric morphometry of the inter-mastoid triangle as a tool for sex determination in the Iranian population. Materials and Methods: The coordinates of 5 landmarks on the mastoid process on the 80 cone-beam computed tomographic images(from individuals aged 17-70 years, 52.5% female) were registered and digitalized. The Cartesian x-y coordinates were acquired for all landmarks, and the shape information was extracted from the principal component scores of generalized Procrustes fit. The t-test was used to compare centroid size. Cross-validated discriminant analysis was used for sex determination. The significance level for all tests was set at 0.05. Results: There was a significant difference in the mastoid size and shape between males and females(P<0.05). The first 2 components of the Procrustes shape coordinates explained 91.3% of the shape variation between the sexes. The accuracy of the discriminant model for sex determination was 88.8%. Conclusion: The application of morphometric geometric techniques will significantly impact forensic studies by providing a comprehensive analysis of differences in biological forms. The results demonstrated that statistical shape analysis can be used as a powerful tool for sex determination based on a morphometric analysis of the inter-mastoid triangle.