Science and Technology Production
Article
Authorship
GALICER, DANIEL ERIC
;
Damián Pinasco
;
Mariano Merzbacher
;
Alexander Litvak
Date
2023
Publishing House and Editing Place
ACADEMIC PRESS INC ELSEVIER SCIENCE
Magazine
JOURNAL OF FUNCTIONAL ANALYSIS
ACADEMIC PRESS INC ELSEVIER SCIENCE
Summary
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We study the volume ratio between emph{projections} of two convex bodies.Given a high-dimensional convex body $K$ we show that there is another convex body$L$ such that the volume ratio between any two projections of fixed rank of the bodies$K$ and $L$ is large. Namely, we prove that for every $1leq kleq n$ and foreach convex body $Ksubset R^n$there is a centrally symmetric body $L subset R^n$ such that for any two projections $P, Q: R^n o R^n$ of rank $k$ one has$$ r(PK, QL) geq c , minleft{ra...
We study the volume ratio between emph{projections} of two convex bodies.Given a high-dimensional convex body $K$ we show that there is another convex body$L$ such that the volume ratio between any two projections of fixed rank of the bodies$K$ and $L$ is large. Namely, we prove that for every $1leq kleq n$ and foreach convex body $Ksubset R^n$there is a centrally symmetric body $L subset R^n$ such that for any two projections $P, Q: R^n o R^n$ of rank $k$ one has$$ r(PK, QL) geq c , minleft{rac{ k}{ sqrt{n}} , sqrt{rac{1}{log log log(rac{nlog(n)}{k})}}, , rac{sqrt{k}}{sqrt{log(rac{nlog(n)}{k})}}},$$ where $c>0$ is an absolute constant.This general lower bound is sharp (up to logarithmic factors)in the regime $kgeq n^{2/3}$.
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Key Words
Projections of convex bodiesVolume ratioConvex BodiesRandom Polytopes