Journal
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
Volume 234, Issue -, Pages -Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.ijsolstr.2021.111276
Keywords
Polyhedron; Static equilibrium; Monostatic polyhedron; Polynomial inequalities; Convex optimization
Categories
Funding
- NKFIH Hungarian Research Fund [134199]
- NKFIH Fund
- Hungarian National Research, Development and Innovation Office (NKFIH) [NKFIA ED_18-2-2018-0006]
- Budapest University of Technology and Economics
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The study focuses on the monostatic property of convex polyhedra, establishing upper and lower bounds for the minimum number of faces and vertices, and improving the lower limit on mono-unstable vertices through an algorithm.
The monostatic property of convex polyhedra (i.e., the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy (1969) published the proof of the existence of the first such object, followed by the constructions of Bezdek (2011) and Reshetov (2014). These examples establish F < 14, V 18 as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy (1969) established for the same problem the lower bounds for the number of faces and vertices as F, V 5 and the same lower bounds were also established for the mono-unstable case (Domokos et al., 2020b). It is also clear that the F, V > 5 bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly one stable and one unstable equilibrium point (both homogeneous and 0-skeletons). In this paper we draw on an unexpected source to extend the knowledge on mono-monostatic solids: we present an algorithm by which we improve the lower bound to V > 8 vertices on mono -unstable 0-skeletons. The problem is transformed into the (un)solvability of systems of polynomial inequalities, which is shown by convex optimization. Our algorithm appears to be less well suited to compute the lower bounds for mono stability. We point out these difficulties in connection with the work of Dawson, Finbow and Mak (Dawson, 1985, Dawson et al., 1998, Dawson and Finbow, 2001) who explored the monostatic property of simplices in higher dimensions.
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