Abstract
We analytically characterize equilateral triangles. Our characterization includes the classically well-known characterizations: for a given triangle in the Euclidean plane, if the centroid and incenter of the triangle coincide, then the triangle must be equilateral; for a given triangle in the Euclidean plane, if the centroid and circumcenter of the triangle coincide, then the triangle must be equilateral. In our characterization, we consider the convolution of a radially symmetric function and the characteristic function of a triangle. The centroid, incenter and circumcenter of a triangle are described in terms of critical points of such a convolution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, H., Kovač, V.: From electrostatic potentials to yet another triangle center. Forum Geom. 15, 73–89 (2015)
Chavel, I., Karp, L.: Movement of hot spots in Riemannian manifolds. J. Anal. Math. 55, 271–286 (1990)
Herburt, I., Moszyńska, M., Peradzyński, Z.: Remarks on radial centres of convex bodies. Math. Phys. Anal. Geom. 8, 157–172 (2005)
Jimbo, S., Sakaguchi, S.: Movement of hot spots over unbounded domains in \({\mathbb{R}}^N\). J. Math. Anal. Appl. 182, 810–835 (1994)
Karp, L., Peyerimhoff, N.: Geometric heat comparison criteria for Riemannian manifolds. Ann. Glob. Anal. Geom. 31, 115–145 (2007)
Kimberling, C.: Encyclopedia of Triangle Centers. Available at https://faculty.evansville.edu/ck6/encyclopedia/etc.html
Magnanini, R., Sakaguchi, S.: The spatial critical points not moving along the heat flow. J. Anal. Math. 71, 237–261 (1997)
Magnanini, R., Sakaguchi, S.: On heat conductors with a stationary hot spot. Ann. Mat. Pura Appl. 183(1), 1–23 (2004)
Moszyńska, M.: Looking for selectors of star bodies. Geom. Dedic. 81, 131–147 (2000)
Moszyńska, M.: Selected Topics in Convex Geometry. Birkhäuser, Boston (2006)
O’Hara, J.: Renormalization of potentials and generalized centers. Adv. Appl. Math. 48, 365–392 (2012)
O’Hara, J., Solanes, G.: Regularized Riesz energies of submanifolds. Math. Nachr. 291, 1356–1373 (2018)
Sakata, S.: Movement of centers with respect to various potentials. Trans. Am. Math. Soc. 367, 8347–8381 (2015)
Sakata, S.: Geometric estimation of a potential and cone conditions of a body. J. Geom. Anal. 27, 2155–2189 (2017)
Sakata, S.: Every convex body has a unique illuminating center. J. Geom. 108, 655–662 (2017)
Sakata, S.: Stationary radial centers and symmetry of convex polytopes. Colloq. Math. 159(1), 91–106 (2020)
Acknowledgements
The author would like to express his deep gratitude to the anonymous reviewer(s) for careful reading(s) and constructive suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is partially supported by JSPS Kakenhi (No. 17K14191), JSPS Overseas Research Fellowships (No. 201860263) and funds (No. 197102) from the Central Research Institute of Fukuoka University.
Rights and permissions
About this article
Cite this article
Sakata, S. Analytic characterization of equilateral triangles. Annali di Matematica 200, 2191–2212 (2021). https://doi.org/10.1007/s10231-021-01075-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-021-01075-9