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Galyna V. Livshyts studies convex geometry, analysis and high-dimensional probability theory. Her research page connects these areas through geometric inequalities and properties of convex sets.^[ref]

Her course Topics in Concentration of Measure and Convexity by Galyna V. Livshyts develops concentration of measure, isoperimetric inequalities and Steiner symmetrization. Replacing each parallel fibre of a set by a centred interval of the same length preserves volume while increasing symmetry, making some inequalities easier to prove. The course page credits the written notes to students, with editing by Livshyts.^[ref]

Steiner symmetrization about a line replaces every perpendicular slice of a measurable set by a centered interval of the same one-dimensional Lebesgue measure. For a bounded Borel set $E\subset {\mathbb{R}}^2$, take the horizontal axis as the symmetry line and putSlice endpoints do not affect area. A slice can consist of several disjoint intervals: its total length determines the replacement.^[ref]

For a convex set bounded by graphs $u_{-}(x)$ and $u_{+}(x)$, the new bounds are $\pm (u_{+}-u_{-})/2$. A disk centered on the symmetry axis stays unchanged. Fubini's theorem proves $\mathrm{area}(E^{\ast })=\int {\ell }_E(x)dx=\mathrm{area}(E)$. Preserved area together with perimeter decrease under Steiner symmetrization makes the operation useful for the isoperimetric inequality.