# The improved isoperimetric inequality and the Wigner caustic of planar ovals

@article{Zwierzynski2016TheII, title={The improved isoperimetric inequality and the Wigner caustic of planar ovals}, author={M. Zwierzy'nski}, journal={Journal of Mathematical Analysis and Applications}, year={2016}, volume={442}, pages={726-739} }

## 16 Citations

A note on Hurwitz's inequality

- Mathematics
- 2017

Abstract Given a simple closed plane curve Γ of length L enclosing a compact convex set K of area F , Hurwitz found an upper bound for the isoperimetric deficit, namely L 2 − 4 π F ≤ π | F e | ,…

The geometry of the Wigner caustic and affine equidistants of planar curves

- Mathematics
- 2016

In this paper we study global properties of the Wigner caustic and affine equidistants of parameterized closed planar curves. We find new results on their geometry and singular points. In particular,…

The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals

- Mathematics
- 2016

In this paper we introduce and study sets: the Constant Width Measure Set and the Spherical Measure Set, which measure the constant width property and the spherical property of planar regular simple…

The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

- MathematicsThe Journal of Geometric Analysis
- 2019

In Saji et al. (J Math 62:259–280, 2008; Ann Math 169:491–529, 2009; J Geom Anal 222):383–409, 2012) the Gauss–Bonnet formulas for coherent tangent bundles over compact-oriented surfaces (without…

The middle hedgehog of a planar convex body

- Mathematics
- 2016

A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in the plane a typical convex body (in the…

Isoperimetric equalities for rosettes

- Mathematics
- 2016

In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented…

The Lower Bounds of the Mixed Isoperimetric Deficit

- Mathematics
- 2021

Let $$K_i$$
be plane convex bodies with the perimeters $$L_{K_i}$$
and areas $$A_{K_i}$$
for $$i=1,2$$
, respectively. In this paper, the lower bounds of the mixed isoperimetric deficit $$\Delta…

Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

- Mathematics, PhysicsBulletin of the Brazilian Mathematical Society, New Series
- 2019

In this paper we study singular points of the Wigner caustic and affine $$\lambda $$ λ -equidistants of planar curves based on shapes of these curves. We generalize the Blaschke–Süss theorem on the…

A note on the isoperimetric deficit

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

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The geometry of the Wigner caustic and affine equidistants of planar curves

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In this paper we study global properties of the Wigner caustic and affine equidistants of parameterized closed planar curves. We find new results on their geometry and singular points. In particular,…

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