M-addition

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• 作者：Tim Mesikepp ^{mesiket@math.washington.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Compact convex set ; Minkowski addition ; LpLp addition ; M-addition ; Valuation ; Shapley&ndash ; Folkman lemma
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：443
• 期：1
• 页码：146-177
• 全文大小：646 K

文摘

M  -addition, denoted by ⊕_M${\oplus }_M$, is a way of combining sets in a vector space which generalizes Minkowski addition +. Under certain circumstances Minkowski addition satisfies (K∩L)+C=(K+C)∩(L+C)$(K\cap L)+C=(K+C)\cap (L+C)$ and (K∪L)+(K∩L)=K+L$(K\cup L)+(K\cap L)=K+L$. We prove parallel properties for M-addition and classify all compact sets M   for which (K∪L)⊕_M(K∩L)=K⊕_ML$(K\cup L){\oplus }_M(K\cap L)=K{\oplus }_{M}L$. Minkowski addition also fulfills $\mathrm{conv}({\sum }_{j=1}^m{A}_j)={\sum }_{j=1}^m\mathrm{conv}{A}_j$, and we classify all sets M   for which $\mathrm{conv}({\oplus }_M(A_1,\dots ,A_m))={\oplus }_{\mathrm{conv}M}(\mathrm{conv}{A}_1,\dots ,\mathrm{conv}{A}_m)$, the natural M  -addition generalization. Corollaries drawn from this result include conditions for when ⊕_M${\oplus }_M$ maps convex polytopes to convex polytopes and an extension of the Shapley–Folkman lemma to M-addition.