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**1 - 4**of**4**### TRISECTIONS OF A 3-ROTATIONALLY SYMMETRIC PLANAR CONVEX BODY MINIMIZING THE MAXIMUM RELATIVE DIAMETER

"... Please cite this article in press as: A. Cañete et al., Trisections of a 3-rotationally symmetric ..."

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Please cite this article in press as: A. Cañete et al., Trisections of a 3-rotationally symmetric

### THE LONGEST SHORTEST FENCE AND SHARP POINCARÉ-SOBOLEV INEQUALITIES

"... Abstract. We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since ..."

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Abstract. We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle. 1.

### On Short Cuts or Fencing in Rectangular Strips

"... Abstract. In this paper we consider an isoperimetric inequality for the free perimeter of a planar shape inside a rectangular domain, the free perimeter being the length of the shape boundary that does not touch the border of the domain. 1 ..."

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Abstract. In this paper we consider an isoperimetric inequality for the free perimeter of a planar shape inside a rectangular domain, the free perimeter being the length of the shape boundary that does not touch the border of the domain. 1