TY - JOUR AU - García-Pacheco, Francisco J. PY - 2012/06/17 Y2 - 2024/03/28 TI - Asymptotically Convex Banach Spaces and The Index of Rotundity Problem JF - Proyecciones (Antofagasta, On line) JA - Proyecciones (Antofagasta, On line) VL - 31 IS - 2 SE - DO - 10.4067/S0716-09172012000200001 UR - https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1147 SP - 91-101 AB - <span style="widows: 2; text-transform: none; text-indent: 0px; display: inline !important; font: 13px Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; float: none; letter-spacing: normal; color: #000000; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">The Index of Rotundity Problem<span class="Apple-converted-space"> </span></span><em style="line-height: normal; widows: 2; text-transform: none; font-variant: normal; text-indent: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; letter-spacing: normal; color: #000000; font-size: 13px; font-weight: normal; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">asks whether a Banach space which admits equivalent renormings with index of rotundity as small as desired also admits an equivalent rotund renorming. In this paper we continue the ongoing search for a negative answer to this question by making use of a new concept:<span class="Apple-converted-space"> </span></em><span style="widows: 2; text-transform: none; text-indent: 0px; display: inline !important; font: 13px Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; float: none; letter-spacing: normal; color: #000000; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">asymptotically convex Banach spaces.<span class="Apple-converted-space"> </span></span><em style="line-height: normal; widows: 2; text-transform: none; font-variant: normal; text-indent: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; letter-spacing: normal; color: #000000; font-size: 13px; font-weight: normal; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">Some applications to<span class="Apple-converted-space"> </span></em><span style="widows: 2; text-transform: none; text-indent: 0px; display: inline !important; font: 13px Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; float: none; letter-spacing: normal; color: #000000; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">The Approximation Hyperplane Series Property<span class="Apple-converted-space"> </span></span><em style="line-height: normal; widows: 2; text-transform: none; font-variant: normal; text-indent: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; white-space: normal; orphans: 2; letter-spacing: normal; color: #000000; font-size: 13px; font-weight: normal; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;">are given.</em> ER -