Seminar:
Analysis
Event time:
Thursday, April 24, 2025 - 4:00pm
Speaker:
Emanuel Milman
Event description:
Let R:L∞(Sn−1)→L∞(Sn−1)R:L∞(Sn−1)→L∞(Sn−1) denote the spherical Radon transform, defined as R(f)(θ)=∫Sn−1∩θ⊥f(u)dσ(u)R(f)(θ)=∫Sn−1∩θ⊥f(u)dσ(u). A long-standing question in non-linear harmonic analysis due to Lutwak, Gardner, and Fish–Nazarov–Ryabogin–Zvavitch, is to characterize those non-negative ρ∈L∞(Sn−1)ρ∈L∞(Sn−1) so that R(ρn−1)=cρR(ρn−1)=cρ when n≥3n≥3. We show that this holds iff ρρ is constant, and moreover, R(R(ρn−1)n−1)=cρR(R(ρn−1)n−1)=cρ iff ρρ is either identically zero or is the reciprocal of some Euclidean norm. Our proof recasts the problem in a geometric language using the intersection body operator II, introduced by Lutwak following the work of Busemann, which plays a central role in the dual Brunn-Minkowski theory. We show that for any star-body KK in RnRn when n≥3n≥3, I2K=cKI2K=cK iff KK is a centered ellipsoid, and hence IK=cKIK=cK iff KK is a centered Euclidean ball. To this end, we interpret the iterated intersection body equation as an Euler-Lagrange equation for a certain volume functional under radial perturbations, derive new formulas for the volume of IKIK, and introduce a continuous version of Steiner symmetrization for Lipschitz star-bodies, which (surprisingly) yields a useful radial perturbation exactly when n≥3n≥3.
Joint work with Shahar Shabelman and Amir Yehudayoff.
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