{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:46:53Z","timestamp":1760230013562,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T00:00:00Z","timestamp":1657152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"specialized research fund of YiBin University","award":["412-2021QH027"],"award-info":[{"award-number":["412-2021QH027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Centroid bodies are a continuous and GL(n)-contravariant valuation and play critical roles in the solution to the Busemann\u2013Petty problem. In this paper, we introduce the notion of harmonic Blaschke\u2013Minkowski homomorphism and show that such a map is represented by a spherical convolution operator. Furthermore, we consider the Shephard-type problem of whether \u03a6K\u2286\u03a6L implies V(K)\u2264V(L), where \u03a6 is a harmonic Blaschke\u2013Minkowski homomorphism. Some important results for centroid bodies are extended to a large class of valuations. Finally, we give two interesting results for even and odd harmonic Blaschke\u2013Minkowski homomorphisms, separately.<\/jats:p>","DOI":"10.3390\/sym14071396","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T22:11:47Z","timestamp":1657231907000},"page":"1396","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Harmonic Blaschke\u2013Minkowski Homomorphism"],"prefix":"10.3390","volume":"14","author":[{"given":"Hongying","family":"Xiao","sequence":"first","affiliation":[{"name":"Faculty of Science, Yibin University, Yibin 644000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weidong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaofeng","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Schneider, R. 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