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In this paper, we study an inverse problem of determining the cross sectionof an infinitely long cylindrical-like material structure from the transverseelectromagnetic scattering measurement. We establish a sharp logarithmicstability result in determining a polygonal scatterer by a single far-fieldmeasurement. The argument in establishing the stability result is localizedaround a corner and can be as well used to produce two highly intriguingimplications for invisibility and transmission resonance in the wave scatteringtheory. In fact, we show that if a generic medium scatterer possesses anadmissible corner on its support, then there exists a positive lower bound ofthe $L^2$-norm of the associated far-field pattern. For the transmissionresonance, we discover a quantitative connection between the regularity of thetransmission eigenfunction at a corner and its analytic or Fourier extension.

Stable determination by a single measurement, scattering bound and regularity of transmission eigenfunction