Convexity in Whitney Problems

Series
Colloquia
Time
Monday, January 27, 2025 - 11:00am for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marjorie Drake – MIT – mkdrake@mit.eduhttps://www.marjoriekdrake.com/
Organizer
Galyna Livshyts

Please Note: This is a job talk, it will be also broadcast by Zoom, in addition to in-person: https://gatech.zoom.us/j/91499035568

Let ERn be a compact set, and f:ER. How can we tell if there exists a smooth convex extension FC1,1(Rn) of f, i.e. satisfying F|E=f|E? Assuming such an extension exists, how small can one take the Lipschitz constant Lip(F):=supx,yRn,xy|F(x)F(y)||xy|? I will provide an answer to these questions for the non-linear space of strongly convex functions by presenting recent work of mine proving there is a Finiteness Principle for strongly convex functions in C1,1(Rn). This work is the first attempt to understand the constrained interpolation problem for *convex* functions in C1,1(Rn), building on techniques developed by P. Shvartsman, C. Fefferman, A. Israel, and K. Luli to understand whether a function has a smooth extension despite obstacles to their direct application. We will finish with a discussion of challenges in adapting my proof of a Finiteness Principle for the space of convex functions in C1,1(R) (n=1) to higher dimensions.