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Motivated by a conjecture of Morales, we study the questions of whether the coefficients of the Ehrhart polynomial of $${\\text {Tes}}_n(1,1,\\dots ,1)$$<\/jats:tex-math>\n \n \n Tes<\/mml:mtext>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are positive. We attack this problem by studying a certain function constructed by Berline\u2013Vergne and its values on faces of a unimodularly equivalent copy of $${\\text {Tes}}_n(1,1,\\dots ,1)$$<\/jats:tex-math>\n \n \n Tes<\/mml:mtext>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We develop a method of obtaining the dot products appeared in formulas for computing Berline\u2013Vergne\u2019s function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline\u2013Vergne\u2019s function has positive values on codimension 2 and\u00a03 faces of the polytopes we consider. As a consequence, we prove that the third and fourth coefficients of the Ehrhart polynomial of $${\\text {Tes}}_{n}(1,\\dots ,1)$$<\/jats:tex-math>\n \n \n Tes<\/mml:mtext>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of $${\\text {Tes}}_{n}(1,\\dots ,1)$$<\/jats:tex-math>\n \n \n Tes<\/mml:mtext>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> including all the integral Tesler polytopes.<\/jats:p>","DOI":"10.1007\/s00454-022-00453-1","type":"journal-article","created":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T19:02:45Z","timestamp":1672340565000},"page":"896-918","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ehrhart Positivity of Tesler Polytopes and Berline\u2013Vergne\u2019s Valuation"],"prefix":"10.1007","volume":"69","author":[{"given":"Yonggyu","family":"Lee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0497-4083","authenticated-orcid":false,"given":"Fu","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,29]]},"reference":[{"issue":"3","key":"453_CR1","first-page":"451","volume":"3","author":"D Armstrong","year":"2012","unstructured":"Armstrong, D., Garsia, A., Haglund, J., Rhoades, B., Sagan, B.: Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics. 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