Convexity of the support of the displacement interpolation: Counterexamples

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• 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 ρ₀,ρ₁${\rho }_0,{\rho }_1$ on two compact convex sets K₀,K₁$K_0,K_1$, respectively, we consider the question whether the support of the measure ρ_t${\rho }_t$ obtained as the geodesic interpolant of ρ₀${\rho }_0$ and ρ₁${\rho }_1$ in the Wasserstein space W₂(R^d) is necessarily convex or not. We prove that this is not the case, even when ρ₀${\rho }_0$ and ρ₁${\rho }_1$ are uniform measures.