Abstract
In this paper we make the final step in finding the optimal way to enclose and separate four planar regions with equal area. In Paolini and Tamagnini (ESAIM COCV 24(3):1303–1331, 2018) the graph-topology of the optimal cluster was found reducing the set of candidates to a one-parameter family of different clusters. With a simple argument we show that the minimal set has a further symmetry and hence is uniquely determined up to isometries.
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Notes
If the three arcs are oriented so that the vertex is the end point of each of the three arcs, then the orientation defines a normal vector \(\mathbf {\nu }\) (for example by rotating the tangent vector counter-clockwise) on the three arcs and the signed curvature k is defined by \(k = \mathbf {k} \cdot \nu \) where \(\mathbf {k}\) is the curvature vector. Clearly k is constant on each circular arc or line segment.
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Communicated by L. Ambrosio.
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