Convexity of the support of the displacement interpolation: Counterexamples

详细信息    查看全文

• 作者：Filippo Santambrogio^a ; ^{filippo.santambrogio@math.u-psud.fr" class="auth_mail" title="E-mail the corresponding author} ; Xu-Jia Wang^b ; ^{Xu-Jia.Wang@anu.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Optimal transport ; Displacement convexity ; Log-concave distributions
• 刊名：Applied Mathematics Letters
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：58
• 期：Complete
• 页码：152-158
• 全文大小：577 K

文摘

Given two smooth and positive densities class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si15.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=334ec4c41b5c4bfbc8c7df0f2ce7b298" title="Click to view the MathML source">ρ₀,ρ₁class="mathContainer hidden">class="mathCode">${\rho }_0,{\rho }_1$ on two compact convex sets class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si16.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=7e19856bcd317242357b933f38fe699c" title="Click to view the MathML source">K₀,K₁class="mathContainer hidden">class="mathCode">$K_0,K_1$, respectively, we consider the question whether the support of the measure class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si17.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=9165cacc63e251f180a17841b80ad73b" title="Click to view the MathML source">ρ_tclass="mathContainer hidden">class="mathCode">${\rho }_t$ obtained as the geodesic interpolant of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si18.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=a9e3066387ecc9f30d1a674d477857a5" title="Click to view the MathML source">ρ₀class="mathContainer hidden">class="mathCode">${\rho }_0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si19.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=bdce340557c6b26a91ddc35f16da91e0" title="Click to view the MathML source">ρ₁class="mathContainer hidden">class="mathCode">${\rho }_1$ in the Wasserstein space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si20.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=b0da6978f99befa0bcc4280a9102606f" title="Click to view the MathML source">W₂(R^d)class="mathContainer hidden">class="mathCode">${\mathbb{W}}_2\left({\mathbb{R}}^d\right)$ is necessarily convex or not. We prove that this is not the case, even when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si18.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=a9e3066387ecc9f30d1a674d477857a5" title="Click to view the MathML source">ρ₀class="mathContainer hidden">class="mathCode">${\rho }_0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916300726&_mathId=si19.gif&_user=111111111&_pii=S0893965916300726&_rdoc=1&_issn=08939659&md5=bdce340557c6b26a91ddc35f16da91e0" title="Click to view the MathML source">ρ₁class="mathContainer hidden">class="mathCode">${\rho }_1$ are uniform measures.