TY - JOUR
T1 - On Aleksandrov-Fenchel Inequalities for k-Convex Domains
AU - Chang, Sun Yung Alice
AU - Wang, Yi
N1 - Funding Information:
The research of the first author is partially supported by NSF grant DMS-0758601.
PY - 2011/6
Y1 - 2011/6
N2 - In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work [9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion [8] for k = 1 case of the inequalities, uses the method of optimal transport.
AB - In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work [9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion [8] for k = 1 case of the inequalities, uses the method of optimal transport.
KW - Aleksandrov-Fenchel inequality
KW - mixed volume
KW - optimal transport map
KW - quermassintegral
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U2 - 10.1007/s00032-011-0159-2
DO - 10.1007/s00032-011-0159-2
M3 - Article
AN - SCOPUS:80051921466
SN - 1424-9286
VL - 79
SP - 13
EP - 38
JO - Milan Journal of Mathematics
JF - Milan Journal of Mathematics
IS - 1
ER -